Solve one-step equations in one variable
Since the 6 is divided, we must multiply it on both sides of the equation.
Solve two-step equations in one variable
Since 7 is subtracted, we must add it to both sides of the equation. Since 2 is being subtracted, we need to add it to both sides of the equation.
Solve multi-step equations involving like-terms
To determine the final solution of the given equation, divide both sides of the equation by 5. Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more to read Click on the ad to read more Click on the ad to read more.
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We should next address the 2 added to 5x by subtracting 2 from both sides of the equation...resulting in the one-step equation: 5x = –35.
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Solve multi-step equations involving parentheses
To determine the final solution of the given equation, divide both sides of the equation by 15. To determine the final solution, divide both sides of the equation by –28 resulting in the solution: x = –3.
Solve multi-step equations where the variable appears on each side of the equation
Since the variable x appears on both sides of the given equation, we must bring one of the variable terms to the other side. Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on ad to read more Click on ad to read more.
Solve multi-step inequalities in one variable
NOTE: Any number value less than or equal to –12 will be considered a correct solution. NOTE: Any number value greater than or equal to 10 will be considered a correct solution.
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Graph the solution of multi-step equations or inequalities (in one variable) on a number line
If the reference value is included within the solution set, we must use a solid dot, and if the reference value is not included in the solution set, we must use an open dot. Since the solution set for the given inequality is x ≠ 3, we must place an open point at +3 and 2 arrows (one to the left and one to the right) from the open point as follows:.
Graph a line that passes through two given points
Connect the two plotted points with a nice long straight line that extends to the outermost part of the coordinate grid.
Graph a linear equation using its x- and y-intercepts
Plot the calculated x-intercept on the coordinate grid by shifting 4 units to the right of the origin (0, 0), but then making no vertical shift. See the blue point located on the right side of the horizontal axis.). Plot the calculated y-intercept on the coordinate grid by not moving horizontally from the origin (0, 0), but then making a vertical shift of 2 units.
Graph a linear equation (in slope-intercept form) using its slope and y-intercept
Use the partial slope to determine the required vertical (rise) and horizontal (run) movements from the y-intercept. Use the partial slope to determine the required vertical. rise) and horizontal (run) moves outside the y boundary.
Graph a linear equation (in standard form) using its slope and y-intercept
Connect the y-intercept to the point determined after the "rise over run" motions (in step #4) with a nice long straight line that extends to the outside of the coordinate grid. The red line is the graph of the given equation passing through the y-intercept and the other point determined by .
Determine the slope of a line passing through 2 points
Note that for chart #3, each successive ordinate (y) has a common difference of zero (0) and each successive abscissa (x) has a common difference of +4. Note that for chart #4, each successive ordinate (y) has a common difference of +4 and each successive abscissa (x) has a common difference of zero (0).
Use the slope-intercept formula to write the equation of a line with a given slope and point
All we have to do is substitute these values into the slope-intercept (y = mx + b) form of the equation to solve for the value of b:. With the value of b = 2 and the value of m = 0, we can write the slope-intercept form of the equation:
Use the point-slope formula to write the slope-intercept form of an equation of a line given its slope and a point
Because we know the slope, we know that m = 0...and because we know a point through which the drawn line will pass, we know that too. Since we know the slope is undefined, we know that the difference between the abscissa is zero (0)... meaning the y-values of all the points shown are identical.
Write the standard form of an equation of a line given its slope and a point
Now we can rearrange the terms so that the x and y terms are on the left side of the equation, while the constant term is on the right side. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. per ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more.
Write the standard form of an equation of a line that passes through 2 given points
Determine whether the lines for a pair of equations are parallel, perpendicular or coincide with each other
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Write the equation of a line through a given point that is parallel to a given line
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Write the equation of a line through a given point that is perpendicular to a given line
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Solve a system of linear equations by graphing
When we solve a system of linear equations, we look for the one point (ordered pair) that will be a solution to all the equations in the system. When we solve a system by graphing, we look for the point of intersection contained in the graphs of the equations in the system.
Solve a system of linear equations by substitution
If this is true, the given system has infinitely many solutions because the lines coincide on a coordinate grid. If this is not true, then the given system has no solution because the lines will be parallel to each other and will not intersect at all.
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Solve a system of linear equations by elimination
After all the variables are gone, we need to determine whether the numerical equation is a true or false statement. Since all the variables have disappeared, this system is a "special" case and we need to determine whether the resulting numerical equation is true or false.
Solve a system of 3 equations in 3 variables
Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. per ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Another adaptation of the method of elimination (or linear combination) is called the right-to-left method, where we first eliminate the rightmost variable and then move to the left ... following the same procedure as the left-to-right method. .
Solve applications involving systems of equations
If the loan for product "A" was for $75,000 more than that for product "B", how much was borrowed for each product.
Find the marginal cost, the marginal cost revenue and the marginal cost profit of given linear total cost functions, linear total revenue
Suppose a stereo receiver manufacturer has the given total cost function and the given total revenue function: C(x) = 200x + 3500 and R(x) = 450x. a) What is the variable cost for this product. Suppose a computer manufacturer has the given total cost function and the given total revenue function: C(x) = 250x + 4000 and R(x) = 750x. a) What is the variable cost for this product.
Evaluate linear total cost functions, linear total revenue functions and linear profit functions
Write the equations for linear total cost functions, linear total revenue functions and linear profit functions by using information given about
Find the break-even point for cost & revenue functions
Evaluate and graph supply & demand functions
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Write the equation (in slope-intercept form) for a supply function and/
To write the equation for the supply function, decide to use the wholesaler theorem. To write the equation for the demand function, determine that the statement about the group of retailers should be used.
Determine the market equilibrium for a given scenario
If we are not sure which given information to use for the supply function or which given information to use for the demand function, just determine the slope for each of the sentences...the negative slope will be the demand function. Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more read Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more.
Graph a linear inequality in two variables
Solve a system of linear inequalities in two variables
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Name the vertices for the feasible solution region for the given system of linear inequalities
Now we need to examine the possible solution region of the given system of linear inequalities to determine its vertices (the points where the sides of the polygonal region intersect). Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more.
Determine the optimal minimum or maximum value of a given linear function that is subject to constraints as defined by the given system
Also, since the first three inequalities (constraints) all use the ratio symbol “≤” (is less than or equal to), all rows should be When the feasible solution region is an "unbounded" region and the shading moves infinitely toward the positive regions of the coordinate plane (as shown in the final graph of Example 1-35b), there will be no maximum value of the given objective function. .
Use graphing to determine the optimal maximum value of a linear function subject to constraints
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Use graphing to determine the optimal minimum value of a linear function subject to constraints
Remember that if we cannot read the vertex coordinates from the grid, we can always determine the intersection points by solving a system of linear equations:
Solve applications involving linear programming
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Determine the number of rows or columns in a matrix
Determine the order (dimensions) of a matrix
Identify square matrices
Identify the element (entry) at a given location (M RC )
Determine a zero matrix with a given order
Find the transpose of a matrix
Determine the negative of a matrix
Use matrices to present and interpret given data
Add matrices
NOTE: As long as the dimensions of the matrices to be added are the same, it makes no difference which of the matrices comes first.
Subtract matrices
NOTE: Since the second matrix in every matrix subtraction becomes its negative, the order in which the problem is presented makes a difference.
Use matrix addition and/or subtraction to solve real-world business applications
Use the appropriate array (matrix) operation to produce the matrix representing the total production at the two plants for September and October by performing the appropriate matrix operation.
Multiply a matrix by a scalar (real number)
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Find the product of given matrices
As seen in Example 2.13e and Example 2.13f, changing the matrices will produce a very different answer. Since both of these representations can be multiplied and matrix multiplication is not normally commutative, it is very important to put the desired matrices in the correct order to achieve the desired product.
Complete matrix multiplication involving zero matrices
Note that whether the null matrix comes first or second, the product is still a null matrix with the same dimensions. Assuming that the presentation of matrix multiplication will allow the multiplication process, multiplication by a zero matrix is an exception to the fact that matrix multiplication is not commutative.
Complete matrix multiplication involving identity matrices
If the identity matrix is the second in the multiplication presentation, its dimensions must match the number of columns in the first matrix. If the identity matrix is the first, its dimensions must match the number of rows in the second matrix.
Use matrix multiplication to solve real-world business applications
We want to show a 25% increase in the entire first column, a 5% increase in the entire second column, a 7.5% increase in the entire third column, a 25% increase in the entire fourth column, an increase of 7 .5% in the entire fifth column and a decrease of 5% in the entire sixth column. Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad for to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more.
Write the augmented matrix for a system of equations
Use a reduced matrix to determine the solutions for a given system
Use matrices to solve systems with unique solutions
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Use matrices to solve systems with non-unique solutions
The goal of this process is to transform the augmented matrix into a "reduced matrix" showing the identity matrix to the left of the vertical line and the correct solutions of the system to the right. Therefore, the third row of the last matrix reads: 0x + 0y + 0z = 0 ... which will work for all values chosen for the variables.
Because the determinant is zero, there is no inverse for this given matrix example 2.21c Find the determinant and inverse of: 1 4.
Use 2 × 2 inverse matrices to solve systems of equations
So it's a good idea not to simplify the fractions inside the inverse before using the inverse in a matrix multiplication.
Use row operations to find the inverse of a square matrix
Whether the system will have no solution or a general solution will depend on the completion of the appropriate row operations required to solve the system. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. per ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad to read more. Click on the ad. for more Click on ad for more Click on ad for more Click on ad for more Click on ad for more Click on ad for more Click on ad for more Click on ad for more Click on ad to read more Click on ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click per ad to read more Click on ad to read more Click on ad to read more Click on ad to read more Click on ad to read more Click on ad to read more Click on ad , to read more Click on the ad for more information.
Use N × N inverse matrices to solve systems of equations
Use matrix inverses to solve real-world applications
Use a computer spreadsheet to determine the number of rows and columns contained in a matrix and thereby determine the order of the
Use a computer to determine the transpose of a matrix
Since this is an array formula, we need to perform a special keystroke combination to enter the formula we want: CRTL+SHIFT+ENTER. Since this is an array formula (array formula), we need to perform the special key combination CRTL+SHIFT+ENTER to enter the desired formula. The formula is surrounded by curly braces, indicating that it is an array formula.) As we can see in the partial screenshot at the bottom right, each of the original inputs has been changed to negative (opposite).
Use a computer spreadsheet to add matrices
Since this is an array (matrix) formula, we will have to do the special CRTL+SHIFT+ENTER keystroke combination to enter the desired formula. Parentheses will surround the formula indicating that it is an array formula.) As we can see from the partial screenshot below on the right, the two entries in respective cells have been subtracted...thus showing the difference of the given matrices.
Use a spreadsheet to complete a scalar multiplication
Use a spreadsheet to complete a matrix multiplication
Use a spreadsheet to solve systems of equations involving row operations
Now that we've swapped rows, we still need to get a 1 in position R2C2 of the desired identity matrix. We have now completed the first and second columns of the desired identity matrix.
Use a spreadsheet to determine the inverse of a matrix
Use a spreadsheet to solve a system of equations involving inverse matrices
Next, we need to highlight another part of the worksheet, which will contain the inverse matrix of coefficients. Remember that to enter the formula we must use the CRTL+SHIFT+ENTER key combination.
Find the greatest common factor of a polynomial
The idea of prime factorization can be extended by introducing variables and exponents into the mix. In fact, when variables and exponents are included, they are already in "prime factorization", since the variable is the base raised to the power (exponent).
Factor a polynomial expression by grouping
To factor this trinomial, we are looking for two factors of –30 that will also add to –1. Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on ad to read more Click on ad to read more Click on ad to read more Click on ad to read more Click on ad to read more Click on ad to read more Click on ad to read more Click on ad to read more Click on ad to read more Click on ad to read more Click on ad to read more Click on ad to read more Click on ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on ad to read more Click on ad to read more Click on ad to read more Click on ad to read more Click on ad to read read more Click on ad to read more Click on ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more.
Since the 36 is positive, the two desired numbers will have the same sign ... and since the 13 is also positive, these signs will be positive. Since the 72 is positive, the two desired numbers will have the same sign ... and since the 18 is negative, these signs will be negative.
Factor “special” polynomials through patterning
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Use factoring to complete business applications
The revenue expression can be factored into two binomials where one of the binomials represents the total number of passengers and the other represents the cost of the boat tour. Factor this given expression to find an expression for the number of units demanded.
Solve quadratic equations by factoring
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Solve quadratic equations by completing the square
Derive the quadratic formula by completing the square
