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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 4: Q 248 (page 458)

Solve system of equations using Cramer鈥檚 Rule.
$\{\begin{cases} - 2 x + 3 y = 3 \\ x + 3 y = 12 \end{cases}$

Short Answer

Expert verified

The solution for the system of linear equation is $(3, 3)$

Step by step solution

01

Step 1. Given information

The system of linear equations is $\{\begin{cases} - 2 x + 3 y = 3 \\ x + 3 y = 12 \end{cases}$

02

Step 2. Find the determinant D and evaluate it

We will take the coefficients of the variables to form the determinant $D$

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

Evaluate it,

$D = (- 2) (3) - (3) (1) D = - 6 - 3 D = - 9$

03

Step 3. Find the determinant Dx and evaluate it

We will use the constants in place of the $x$ coefficients to find the determinant $D_{x}$

$D_{x} = |\begin{matrix} 3 & 3 \\ 12 & 3 \end{matrix}|$

Evaluate it

$D_{x} = (3) (3) - (3) (12) D_{x} = 9 - 36 D_{x} = - 27$

04

Step 4. Find the determinant Dy and evaluate it

We will use the constants in place of the $y$ coefficients to find the determinant $D_{y}$

$D_{y} = |\begin{matrix} - 2 & 3 \\ 1 & 12 \end{matrix}|$

Evaluate it

$D_{y} = (- 2) (12) - (3) (1) D_{y} = - 24 - 3 D_{y} = - 27$

05

Step 5. Find x and y

To find $x$ ,

$x = \frac{D_{x}}{D} x = \frac{- 27}{- 9} x = 3$

To find $y$ ,

$y = \frac{D_{y}}{D} y = \frac{- 27}{- 9} y = 3$

The solution for the system of linear equation is $(3, 3)$

06

Step 6. Check by substituting values

Substitute $x = 3, y = 3$ in both the equations,

$x + 3 y = 12 3 + 3 (3) = 12 3 + 9 = 12 12 = 12$

This is true.

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

This is also true.

Therefore, the solution is $(3, 3)$

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Most popular questions from this chapter

The sum of two number is

$- 26$ . One number is $12$ less than

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\{\begin{cases} 3 x - 4 y = - 9 \\ 5 x + 3 y = 14 \end{cases}

Translate to a system of equations and solve:

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$\{\begin{cases} 2 x + y = 5 \\ x - 2 y = - 15 \end{cases}$

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