Question

how to solve for x in an problem with 2 equations

Answer 1

To solve for \( x \) in a system of two equations, you can use various methods, including substitution, elimination, or graphical methods. Here’s a step-by-step guide for each method:

### 1. Substitution Method

**Step 1:** Solve one equation for one variable.

Suppose you have the following system:
\[
\begin{align*}
1) & \quad 2x + 3y = 6 \\
2) & \quad x - y = 4
\end{align*}
\]

From the second equation, solve for \( x \):
\[
x = y + 4
\]

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

Replace \( x \) in the first equation:
\[
2(y + 4) + 3y = 6
\]
\[
2y + 8 + 3y = 6
\]
\[
5y + 8 = 6
\]
\[
5y = -2
\]
\[
y = -\frac{2}{5}
\]

**Step 3:** Substitute back to find \( x \).

Using \( y = -\frac{2}{5} \) in \( x = y + 4 \):
\[
x = -\frac{2}{5} + 4 = \frac{18}{5}
\]

**Final Solutions:**
\[
x = \frac{18}{5}, \quad y = -\frac{2}{5}
\]

### 2. Elimination Method

**Step 1:** Align the equations.

\[
\begin{align*}
1) & \quad 2x + 3y = 6 \\
2) & \quad x - y = 4
\end{align*}
\]

**Step 2:** Make coefficients of one variable equal (if needed).

Multiply the second equation by 2:
\[
\begin{align*}
1) & \quad 2x + 3y = 6 \\
2) & \quad 2x - 2y = 8
\end{align*}
\]

**Step 3:** Subtract one equation from the other.

Subtract the first equation from the second:
\[
(2x - 2y) - (2x + 3y) = 8 - 6
\]
\[
-5y = 2
\]
\[
y = -\frac{2}{5}
\]

**Step 4:** Substitute back to find \( x \).

Using \( y = -\frac{2}{5} \) in the second original equation:
\[
x - \left(-\frac{2}{5}\right) = 4 \implies x + \frac{2}{5} = 4
\]
\[
x = 4 - \frac{2}{5} = \frac{18}{5}
\]

**Final Solutions:**
\[
x = \frac{18}{5}, \quad y = -\frac{2}{5}
\]

### 3. Graphical Method

**Step 1:** Plot each equation on a graph.

Convert each equation into slope-intercept form ( \( y = mx + b \) ).

**Step 2:** Identify the point of intersection.

The point where the two lines cross is the solution \((x, y)\).

### Summary

Choose a method you are most comfortable with. After going through these steps, you will have the values of \( x \) and \( y \) that satisfy both equations. If you'd like help with specific equations, feel free to provide them!