Let's solve the equation step-by-step to find the values of \(x\).
First, we have the two equations:
[tex]\[
y_1 = 3(5x - 4) - 7
\][/tex]
[tex]\[
y_2 = 13(x - 4) + 49
\][/tex]
We are given that \(y_1 = y_2\). So, we set these two equations equal to each other:
[tex]\[
3(5x - 4) - 7 = 13(x - 4) + 49
\][/tex]
Next, we need to simplify both sides of the equation:
Starting with the left-hand side,
[tex]\[
3(5x - 4) - 7
\][/tex]
We first distribute the 3:
[tex]\[
3 \cdot 5x - 3 \cdot 4 - 7
\][/tex]
[tex]\[
15x - 12 - 7
\][/tex]
[tex]\[
15x - 19
\][/tex]
Now, for the right-hand side,
[tex]\[
13(x - 4) + 49
\][/tex]
We distribute the 13:
[tex]\[
13 \cdot x - 13 \cdot 4 + 49
\][/tex]
[tex]\[
13x - 52 + 49
\][/tex]
[tex]\[
13x - 3
\][/tex]
With both sides simplified, we have:
[tex]\[
15x - 19 = 13x - 3
\][/tex]
To isolate \(x\), we first move the \(x\) terms to one side and the constant terms to the other side. Subtract \(13x\) from both sides:
[tex]\[
15x - 13x - 19 = -3
\][/tex]
[tex]\[
2x - 19 = -3
\][/tex]
Next, add 19 to both sides:
[tex]\[
2x = -3 + 19
\][/tex]
[tex]\[
2x = 16
\][/tex]
Finally, divide both sides by 2:
[tex]\[
x = \frac{16}{2}
\][/tex]
[tex]\[
x = 8
\][/tex]
Therefore, the value of \(x\) that satisfies the given conditions is:
[tex]\[
x = 8
\][/tex]
