To solve the system of equations by eliminating the [tex]\(x\)[/tex]-terms, we can use the method of elimination. Here are the steps to follow:
1. System of Equations:
[tex]\[
\begin{array}{l}
6x - 2y = 2 \quad \quad \quad \quad \quad \quad (1) \\
8x + 3y = 14 \quad \quad \quad \quad \quad (2)
\end{array}
\][/tex]
2. Find the Least Common Multiple (LCM) of the coefficients of [tex]\(x\)[/tex]:
The coefficients of [tex]\(x\)[/tex] in the two equations are 6 and 8. To eliminate the [tex]\(x\)[/tex]-terms, we need to make their coefficients the same in both equations. This is done by finding the LCM of these coefficients.
3. LCM Calculation:
The Least Common Multiple (LCM) of 6 and 8 is 24.
4. Making Coefficients of [tex]\(x\)[/tex] Equal to LCM:
- Multiply Equation (1) by 4 (since LCM / 6 = 24 / 6 = 4):
[tex]\[
4 \times (6x - 2y) = 4 \times 2
\][/tex]
This simplifies to:
[tex]\[
24x - 8y = 8 \quad \quad \quad \quad \quad \quad (3)
\][/tex]
- Multiply Equation (2) by 3 (since LCM / 8 = 24 / 8 = 3):
[tex]\[
3 \times (8x + 3y) = 3 \times 14
\][/tex]
This simplifies to:
[tex]\[
24x + 9y = 42 \quad \quad \quad \quad \quad \quad (4)
\][/tex]
5. Subtract the New Equations to Eliminate [tex]\(x\)[/tex]:
Now, subtract Equation (3) from Equation (4) to eliminate the [tex]\(x\)[/tex]-terms:
[tex]\[
(24x + 9y) - (24x - 8y) = 42 - 8
\][/tex]
This simplifies to:
[tex]\[
17y = 34
\][/tex]
6. Solve for [tex]\(y\)[/tex]:
Divide both sides by 17:
[tex]\[
y = \frac{34}{17} = 2
\][/tex]
7. Substitute [tex]\(y\)[/tex] back into one of the original equations to solve for [tex]\(x\)[/tex]:
Use Equation (1):
[tex]\[
6x - 2(2) = 2
\][/tex]
Simplify the equation:
[tex]\[
6x - 4 = 2
\][/tex]
Add 4 to both sides:
[tex]\[
6x = 6
\][/tex]
Divide by 6:
[tex]\[
x = 1
\][/tex]
The solution to the system of equations is:
[tex]\[
x = 1, \quad y = 2
\][/tex]
