To solve the system of simultaneous equations:
[tex]\[
\begin{cases}
2x + 3y = 13 \\
4x - y = -2
\end{cases}
\][/tex]
we can follow these steps:
### Step 1: Solve one of the equations for one variable
Let's solve the second equation for \(y\):
[tex]\[
4x - y = -2
\][/tex]
Isolating \(y\):
[tex]\[
y = 4x + 2
\][/tex]
### Step 2: Substitute the expression for \(y\) into the first equation
Take the expression for \(y\) from the second equation and substitute it into the first equation:
[tex]\[
2x + 3(4x + 2) = 13
\][/tex]
### Step 3: Simplify the substituted equation and solve for \(x\)
Distribute and simplify:
[tex]\[
2x + 12x + 6 = 13
\][/tex]
Combine like terms:
[tex]\[
14x + 6 = 13
\][/tex]
Subtract 6 from both sides:
[tex]\[
14x = 7
\][/tex]
Divide both sides by 14:
[tex]\[
x = \frac{7}{14} = \frac{1}{2}
\][/tex]
### Step 4: Substitute the value of \(x\) back into the expression for \(y\)
Now substitute \(x = \frac{1}{2}\) back into the expression for \(y\):
[tex]\[
y = 4 \left( \frac{1}{2} \right) + 2
\][/tex]
Multiply:
[tex]\[
y = 2 + 2 = 4
\][/tex]
### Step 5: Verify the solution
It's always a good idea to verify the solution by plugging \(x = \frac{1}{2}\) and \(y = 4\) back into the original equations to ensure they hold true:
For the first equation:
[tex]\[
2 \left( \frac{1}{2} \right) + 3(4) = 1 + 12 = 13
\][/tex]
For the second equation:
[tex]\[
4 \left( \frac{1}{2} \right) - 4 = 2 - 4 = -2
\][/tex]
Both equations are satisfied.
Therefore, the solution to the system of equations is:
[tex]\[
x = \frac{1}{2}, \quad y = 4
\][/tex]
