Question

Use the substitution method to solve the system
𝑥 − 2𝑦 = 3 & 4𝑥 − 8𝑦 = 12

Answer 1

To solve the system of equations using the substitution method, we start with the two equations given:

1. \( x - 2y = 3 \)
2. \( 4x - 8y = 12 \)

First, we'll solve the first equation for \( x \):

\[
x = 2y + 3
\]

Now that we have \( x \) expressed in terms of \( y \), we can substitute this expression for \( x \) into the second equation:

\[
4(2y + 3) - 8y = 12
\]

Now, distribute the 4:

\[
8y + 12 - 8y = 12
\]

The \( 8y \) terms cancel each other out:

\[
12 = 12
\]

This statement is always true, which indicates that the two equations are actually equivalent. This means that the system has infinitely many solutions, as they represent the same line.

To express the solution, we can write:

- The variable \( y \) can be any real number \( t \).
- Substituting \( t \) into the expression for \( x \):

\[
x = 2t + 3
\]

Thus, the solutions can be expressed as:

\[
(x, y) = (2t + 3, t)
\]

where \( t \) is any real number. Therefore, the solution set is the line defined by:

\[
y = \frac{1}{2}(x - 3)
\]

In summary, there are infinitely many solutions to the system, represented by the equations of the line.