To solve the system of equations given, we will use a step-by-step approach. The system of equations is:
1. [tex]\(12x + 8y = 4\)[/tex]
2. [tex]\(18x + 10y = 7\)[/tex]
We need to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations.
### Step-by-Step Solution:
1. Define the equations:
[tex]\[
12x + 8y = 4 \quad \text{(Equation 1)}
\][/tex]
[tex]\[
18x + 10y = 7 \quad \text{(Equation 2)}
\][/tex]
2. Simplify if possible:
- Equation 1: [tex]\( 12x + 8y = 4 \)[/tex]
- Equation 2 cannot be simplified in a straightforward manner without altering the given coefficients significantly.
3. Solving using a method of elimination or substitution:
- Let's use the method of elimination:
- Multiply Equation 1 by [tex]\(18\)[/tex] and Equation 2 by [tex]\(12\)[/tex] to make the coefficients of [tex]\(x\)[/tex] the same:
[tex]\[
18(12x + 8y) = 18(4) \implies 216x + 144y = 72 \quad \text{(Transformed Equation 1)}
\][/tex]
[tex]\[
12(18x + 10y) = 12(7) \implies 216x + 120y = 84 \quad \text{(Transformed Equation 2)}
\][/tex]
4. Subtract the transformed equations to eliminate [tex]\(x\)[/tex]:
[tex]\[
(216x + 144y) - (216x + 120y) = 72 - 84
\][/tex]
[tex]\[
216x + 144y - 216x - 120y = -12
\][/tex]
[tex]\[
24y = -12
\][/tex]
[tex]\[
y = -\frac{12}{24} = -\frac{1}{2}
\][/tex]
5. Substitute the value of [tex]\(y\)[/tex] back into one of the original equations to find [tex]\(x\)[/tex]:
- Substitute [tex]\( y = -\frac{1}{2} \)[/tex] into Equation 1:
[tex]\[
12x + 8\left(-\frac{1}{2}\right) = 4
\][/tex]
[tex]\[
12x - 4 = 4
\][/tex]
[tex]\[
12x = 8
\][/tex]
[tex]\[
x = \frac{8}{12} = \frac{2}{3}
\][/tex]
### Conclusion:
The solution to the system of equations is:
[tex]\[
x = \frac{2}{3}, \quad y = -\frac{1}{2}
\][/tex]
Therefore, the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both given equations are:
[tex]\[
\left(\frac{2}{3}, -\frac{1}{2}\right)
\][/tex]
