Question

(a)
Solve the system of equations using elimination.

−
2
x
+
6
y
=
−
4
−
4
x
+
12
y
=
−
8
−2x+6y=−4
−4x+12y=−8

Answer 1

To solve the system of equations using the elimination method, we first write down the equations:

1. \(-2x + 6y = -4\) (Equation 1)
2. \(-4x + 12y = -8\) (Equation 2)

Next, we notice that Equation 2 is a multiple of Equation 1. Specifically, if we multiply Equation 1 by 2, we get:

\[
2(-2x + 6y) = 2(-4)
\]
\[
-4x + 12y = -8
\]

This shows that Equation 2 is indeed the same as Equation 1, meaning they represent the same line in the coordinate plane. Therefore, there are infinitely many solutions along that line.

To express this line, we can rearrange Equation 1 to solve for \(y\):

Starting with:
\[
-2x + 6y = -4
\]

Add \(2x\) to both sides:
\[
6y = 2x - 4
\]

Now divide by 6:
\[
y = \frac{1}{3}x - \frac{2}{3}
\]

Thus, the solution to the system can be expressed as:
\[
y = \frac{1}{3}x - \frac{2}{3}
\]

This means any pair \((x, y)\) that satisfies this equation is a solution to the original system. For example, if \(x = 0\), then \(y = -\frac{2}{3}\), which gives the solution \((0, -\frac{2}{3})\).

In conclusion, the system has infinitely many solutions along the line represented by:
\[
y = \frac{1}{3}x - \frac{2}{3}
\]