Use substutution to solve the system. $$ \begin{array}{r}-2 x+3 y=6 \ 3 x-5=y\end{array} $$

Question

UpStudy AI · Accepted Answer

Sure, let's solve the given system of equations using the **substitution method**.

The system is:

$$
\begin{cases}
-2x + 3y = 6 \quad \text{(1)} \
3x - 5 = y \quad \text{(2)}
\end{cases}
$$

**Step 1: Solve one of the equations for one variable**

From equation (2):

$$
y = 3x - 5
$$

**Step 2: Substitute this expression into the other equation**

Plug the expression for $$ y $$ from equation (2) into equation (1):

$$
-2x + 3(3x - 5) = 6
$$

**Step 3: Simplify and solve for $$ x $$**

$$
-2x + 9x - 15 = 6
$$

Combine like terms:

$$
7x - 15 = 6
$$

Add 15 to both sides:

$$
7x = 21
$$

Divide both sides by 7:

$$
x = 3
$$

**Step 4: Substitute $$ x = 3 $$ back into equation (2) to find $$ y $$**

$$
y = 3(3) - 5 = 9 - 5 = 4
$$

**Solution:**

$$
x = 3,\quad y = 4
$$

**Verification:**

Let's verify the solution by plugging $$ x = 3 $$ and $$ y = 4 $$ back into both original equations.

1. **First equation:**

$$
-2(3) + 3(4) = -6 + 12 = 6 \quad \text{✓}
$$

2. **Second equation:**

$$
3(3) - 5 = 9 - 5 = 4 \quad \text{✓}
$$

Both equations are satisfied, so the solution is correct.

**Final Answer:**

$$
\boxed{ \begin{cases} x = 3 \ y = 4 \end{cases} }
$$
The solution is $$ x = 3 $$ and $$ y = 4 $$.

Use substutution to solve the system. \[ \begin{array}{r}-2 x+3 y=6 \\ 3 x-5=y\end{array} \]