Question

how to solve for x in an problem with 2 equations and a equal sign between them

Answer 1

To solve for \( x \) when you have two equations set equal to each other, you can follow these general steps:

1. **Identify the equations**: Write down both equations clearly. For example, let’s say you have:

\[
A(x) = B(x)
\]

where \( A(x) \) and \( B(x) \) are expressions involving \( x \).

2. **Set the equations equal**: Since you have \( A(x) = B(x) \), you can rearrange the equation to set the equation to zero. For instance:

\[
A(x) - B(x) = 0
\]

3. **Combine like terms**: Simplify the equation from step 2 to combine like terms where possible.

4. **Isolate \( x \)**: If the equation is linear, you can solve for \( x \) by moving terms around to isolate \( x \). For example, if you end up with a linear equation like:

\[
mx + b = 0
\]

you can isolate \( x \) by subtracting \( b \) from both sides and then dividing by \( m \).

5. **Check for solutions**: If the equation is quadratic or of higher degree, you may need to factor, use the quadratic formula, or other methods appropriate to the type of equation you end up with.

6. **Verify the solution**: Once you've found a solution for \( x \), it's always a good idea to substitute it back into the original equations to ensure that both sides are equal.

**Example**:

Consider the equations:

\[
2x + 3 = x + 6
\]

1. Set the equations equal and rearrange:

\[
2x + 3 - (x + 6) = 0
\]

which simplifies to:

\[
2x + 3 - x - 6 = 0
\]

or

\[
x - 3 = 0
\]

2. Solve for \( x \):

\[
x = 3
\]

3. Verify:

Substitute \( x = 3 \) back into the original equations:

- Left side: \( 2(3) + 3 = 6 + 3 = 9 \)
- Right side: \( 3 + 6 = 9 \)

Since both sides are equal, \( x = 3 \) is the correct solution.