Question

Solve the system of equations by substitution

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

Short answer

The solution for the system of equations is,

$$\begin{gathered} x=3 \\ y=-1 \end{gathered}$$.

Step-by-step solution

Step 1. Given information

The system of equations is,

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

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

Substitute $y=\frac{1}{3}x-2$in the equation

$2x+5y=1$

$2x+5(\frac{1}{3}x-2)=1$

role="math" localid="1644341510028" $2x+\frac{5}{3}x-10=1$

$2x+\frac{5x}{3}=1+10$

$\frac{6x+5x}{3}=11$

$\frac{11x}{3}=11$

$x=11÷\frac{11}{3}$

$x=11\times \frac{3}{11}$

$x=3$

Step 3. Substituting the value of x in the equation

Substituting the value of $x=3$in the equation

$y=\frac{1}{3}x-2$

$y=\frac{1}{3}\left(3\right)-2$

$y=1-2$

$y=-1$

Step 4. Checking the solution for the equation 2x+5y=1

Substituting $$\begin{gathered} x=3 \\ y=-1 \end{gathered}$$in the equation

$2x+5y=1$

$2\left(3\right)+5(-1)=1$

$6+(-5)=1$

$6-5=1$

$1=1$

This is true.

Step 5. Checking the solution for the equation y=13x-2

Substituting $$\begin{gathered} x=3 \\ y=-1 \end{gathered}$$in the equation

$y=\frac{1}{3}x-2$

$-1=\frac{1}{3}\left(3\right)-2$

$-1=1-2$

$-1=-1$

This is true.