Linear Algebra Examples

$x = 7 (y - 5)$ , $4 y = 3 x + 5$

Step 1

Move variables to the left and constant terms to the right.

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Step 1.1

Subtract $7 (y - 5)$ from both sides of the equation.

$x - 7 (y - 5) = 0$

$4 y = 3 x + 5$

Step 1.2

Simplify each term.

$x - 7 y + 35 = 0$

$4 y = 3 x + 5$

Step 1.3

Subtract $35$ from both sides of the equation.

$x - 7 y = - 35$

$4 y = 3 x + 5$

Step 1.4

Subtract $3 x$ from both sides of the equation.

$x - 7 y = - 35$

$4 y - 3 x = 5$

Step 1.5

Reorder $4 y$ and $- 3 x$ .

$x - 7 y = - 35$

$- 3 x + 4 y = 5$

$x - 7 y = - 35$

$- 3 x + 4 y = 5$

Step 2

Write the system as a matrix.

$[\begin{array}{c} 1 & - 7 & - 35 \\ - 3 & 4 & 5 \end{array}]$

Step 3

Find the reduced row echelon form.

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Step 3.1

Perform the row operation $R_{2} = R_{2} + 3 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 3.1.1

Perform the row operation $R_{2} = R_{2} + 3 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - 7 & - 35 \\ - 3 + 3 \cdot 1 & 4 + 3 \cdot - 7 & 5 + 3 \cdot - 35 \end{array}]$

Step 3.1.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 7 & - 35 \\ 0 & - 17 & - 100 \end{array}]$

Step 3.2

Multiply each element of $R_{2}$ by $- \frac{1}{17}$ to make the entry at $2, 2$ a $1$ .

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Step 3.2.1

Multiply each element of $R_{2}$ by $- \frac{1}{17}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 7 & - 35 \\ - \frac{1}{17} \cdot 0 & - \frac{1}{17} \cdot - 17 & - \frac{1}{17} \cdot - 100 \end{array}]$

Step 3.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 7 & - 35 \\ 0 & 1 & \frac{100}{17} \end{array}]$

Step 3.3

Perform the row operation $R_{1} = R_{1} + 7 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 3.3.1

Perform the row operation $R_{1} = R_{1} + 7 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + 7 \cdot 0 & - 7 + 7 \cdot 1 & - 35 + 7 (\frac{100}{17}) \\ 0 & 1 & \frac{100}{17} \end{array}]$

Step 3.3.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & \frac{105}{17} \\ 0 & 1 & \frac{100}{17} \end{array}]$

Step 4

Use the result matrix to declare the final solution to the system of equations.

$x = \frac{105}{17}$

$y = \frac{100}{17}$

Step 5

The solution is the set of ordered pairs that make the system true.

$(\frac{105}{17}, \frac{100}{17})$