Chapter 4: Q. 391 (page 483)

In the following exercises, solve each system of equations using Cramer’s rule
$\{\begin{cases} x - 3 y = - 9 \\ 2 x + 5 y = 4 \end{cases}$

Short Answer

Expert verified

The ordered pair $(- 3, 2)$ is the solution for the given system of equations.

Step by step solution

Step 1. Given information

The given system of equations is:

$\{\begin{cases} x - 3 y = - 9 \\ 2 x + 5 y = 4 \end{cases}$

Step 2. Evaluating the determinant D

In determinant $D$ all the coefficients are taken.

So,

$D = |\begin{matrix} 1 & - 3 \\ 2 & 5 \end{matrix}|$

$D = 1 \times 5 - (- 3 \times 2) D = 5 + 6 D = 11$

Step 3. Evaluating the determinant Dx

In determinant $D_{x}$ , we take the constants in place of coefficients of $x$

So.

$D_{x} = |\begin{matrix} - 9 & - 3 \\ 4 & 5 \end{matrix}| D_{x} = - 9 \times 5 - (- 3 \times 4) D_{x} = - 45 + 12 D_{x} = - 33$

Step 4. Evaluating the determinant Dy

In the determinant $D_{y}$ , we take the constant in place of coefficients of $y$

So,

$D_{y} = |\begin{matrix} 1 & - 9 \\ 2 & 4 \end{matrix}| D_{y} = 1 \times 4 - (- 9 \times 2) D_{y} = 4 + 18 D_{y} = 22$

Step 5. Finding x and y

For $x$

$x = \frac{D_{x}}{D} x = \frac{- 33}{11} x = - 3$

For $y$

$y = \frac{D_{y}}{D} y = \frac{22}{11} y = 2$

Step 6. Writing solution in ordered pair

The solution of the system in ordered pair is $(- 3, 2)$

Step 7. Check the solution for x − 3y = −9

Substituting $x = - 3 y = 2$ in the equation, we get:

$x - 3 y = - 9 - 3 - 3 (2) = - 9 - 3 - 6 = - 9 - 9 = - 9$

This is true

Step 8. Check the solution for 2x + 5y = 4

Substituting $x = - 3 y = 2$ in the equation, we get:

$2 x + 5 y = 4 2 (- 3) + 5 (2) = 4 - 6 + 10 = 4 4 = 4$

This is also true.

Thus, the ordered pair $(- 3, 2)$ is the solution.

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In the following exercises, solve each system of equations using Cramer’s rule
$\{\begin{cases} x - 3 y = - 9 \\ 2 x + 5 y = 4 \end{cases}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Evaluating the determinant D

Step 3. Evaluating the determinant Dx

Step 4. Evaluating the determinant Dy

Step 5. Finding x and y

Step 6. Writing solution in ordered pair

Step 7. Check the solution for x − 3y = −9

Step 8. Check the solution for 2x + 5y = 4

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In the following exercises, solve each system of equations using Cramer’s rulex-3y=-92x+5y=4

Short Answer

Step by step solution

Step 1. Given information

Step 2. Evaluating the determinant D

Step 3. Evaluating the determinant Dx

Step 4. Evaluating the determinant Dy

Step 5. Finding x and y

Step 6. Writing solution in ordered pair

Step 7. Check the solution for x − 3y = −9

Step 8. Check the solution for 2x + 5y = 4

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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In the following exercises, solve each system of equations using Cramer’s rule
$\{\begin{cases} x - 3 y = - 9 \\ 2 x + 5 y = 4 \end{cases}$