Solve the system by substitution. $$ \begin{aligned} y & =5 x+30 \ y & =2 x\end{aligned} $$

Question

UpStudy AI · Accepted Answer

To solve the system of equations using substitution, follow these steps:

$$
\begin{aligned}
1) \quad y &= 5x + 30 \
2) \quad y &= 2x
\end{aligned}
$$

**Step 1: Substitute Equation (2) into Equation (1)**

Since both equations equal $$ y $$, you can set them equal to each other:

$$
5x + 30 = 2x
$$

**Step 2: Solve for $$ x $$**

Subtract $$ 2x $$ from both sides to get all $$ x $$-terms on one side:

$$
5x - 2x + 30 = 0
$$
$$
3x + 30 = 0
$$

Subtract 30 from both sides:

$$
3x = -30
$$

Divide both sides by 3:

$$
x = -10
$$

**Step 3: Substitute $$ x $$ back into one of the original equations to find $$ y $$**

Using Equation (2):

$$
y = 2x = 2(-10) = -20
$$

**Solution:**

$$
x = -10 \
y = -20
$$

**Verification:**

You can verify the solution by plugging the values back into both original equations.

- For Equation (1):
  $$
  y = 5(-10) + 30 = -50 + 30 = -20
  $$
  
- For Equation (2):
  $$
  y = 2(-10) = -20
  $$

Both equations are satisfied, so $$ x = -10 $$ and $$ y = -20 $$ is the correct solution.
$$ x = -10 $$ and $$ y = -20 $$.

Solve the system by substitution. \[ \begin{aligned} y & =5 x+30 \\ y & =2 x\end{aligned} \]