Question

Solve the system of equations by substitution

$\left\{\begin{array}{l}3x+4y=1\\ y=-\frac{2}{5}x+2\end{array}\right.$

Short answer

The solution for the system of equations is,

$$\begin{gathered} x=-5 \\ y=4 \end{gathered}$$

Step-by-step solution

Step 1. Given the information

The system of equations is,

$\left\{\begin{array}{l}3x+4y=1\\ y=-\frac{2}{5}x+2\end{array}\right.$

Step 2. Substituting the value of y in the equation.

Substituting $y=-\frac{2}{5}x+2$in the equation,

$3x+4y=1$

role="math" localid="1644342937459" $3x+4(-\frac{2}{5}x+2)=1$

$3x-\frac{8}{5}x+8=1$

$3x-\frac{8x}{5}=1-8$

$\frac{15x-8x}{5}=-7$

$\frac{7x}{5}=-7$

$x=-7÷\frac{7}{5}$

$x=-7\times \frac{5}{7}$

$x=-5$

Step 3. Substituting the value of x in the equation

Substituting $x=-5$in the equation

$y=-\frac{2}{5}x+2$

$y=-\frac{2}{5}(-5)+2$

$y=2+2$

$y=4$

Step 4. Checking the solution for the equation 3x+4y=1

Substituting $$\begin{gathered} x=-5 \\ y=4 \end{gathered}$$in the equation

$3x+4y=1$

$3(-5)+4\left(4\right)=1$

$-15+16=1$

$1=1$

This is true

Step 5. Checking the solution for the equation y=-25x+2

Substituting $$\begin{gathered} x=-5 \\ y=4 \end{gathered}$$in the equation

$y=-\frac{2}{5}x+2$

$4=-\frac{2}{5}(-5)+2$

$4=2+2$

$4=4$

This is true.