Chapter 4: Q 248 (page 458)

Solve system of equations using Cramer’s 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

Step 1. Given information

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

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,

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

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

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

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

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

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 $(3, 3)$

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

Short Answer

Step by step solution

Step 1. Given information

Step 2. Find the determinant D and evaluate it

Step 3. Find the determinant Dx and evaluate it

Step 4. Find the determinant Dy and evaluate it

Step 5. Find x and y

Step 6. Check by substituting values

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Solve system of equations using Cramer’s Rule.-2x+3y=3x+3y=12

Short Answer

Step by step solution

Step 1. Given information

Step 2. Find the determinant D and evaluate it

Step 3. Find the determinant Dx and evaluate it

Step 4. Find the determinant Dy and evaluate it

Step 5. Find x and y

Step 6. Check by substituting values

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Geometry

Logic and Functions

Mechanics Maths

Pure Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

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