College

Michael graphs the equations [tex]y=-\frac{1}{2}x+4[/tex] and [tex]y=x+1[/tex] to solve the equation [tex]-\frac{1}{2}x+4=x+1[/tex].

What are the solution(s) of [tex]-\frac{1}{2}x+4=x+1[/tex]?

Answer :

To solve the equation [tex]\(-\frac{1}{2}x + 4 = x + 1\)[/tex], we need to find the value of [tex]\(x\)[/tex] that makes both sides equal.

Here are the steps to solve the equation:

1. Move the terms involving [tex]\(x\)[/tex] to one side of the equation:

Start by adding [tex]\(\frac{1}{2}x\)[/tex] to both sides to eliminate the [tex]\(x\)[/tex]-term from the left side. The equation becomes:
[tex]\[
4 = x + \frac{1}{2}x + 1
\][/tex]

2. Combine like terms on the right side:

Combine [tex]\(x\)[/tex] and [tex]\(\frac{1}{2}x\)[/tex] on the right side:
[tex]\[
x + \frac{1}{2}x = \frac{3}{2}x
\][/tex]
So now, the equation is:
[tex]\[
4 = \frac{3}{2}x + 1
\][/tex]

3. Isolate the [tex]\(x\)[/tex]-term:

Subtract 1 from both sides to isolate the [tex]\(\frac{3}{2}x\)[/tex] term on the right:
[tex]\[
4 - 1 = \frac{3}{2}x
\][/tex]
[tex]\[
3 = \frac{3}{2}x
\][/tex]

4. Solve for [tex]\(x\)[/tex]:

To isolate [tex]\(x\)[/tex], divide both sides by [tex]\(\frac{3}{2}\)[/tex]:
[tex]\[
x = \frac{3}{3/2}
\][/tex]

5. Simplify the division:

Dividing 3 by [tex]\(\frac{3}{2}\)[/tex] is the same as multiplying 3 by the reciprocal of [tex]\(\frac{3}{2}\)[/tex], which is [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[
x = 3 \times \frac{2}{3} = 2
\][/tex]

So, the solution to the equation [tex]\(-\frac{1}{2}x + 4 = x + 1\)[/tex] is [tex]\(x = 2\)[/tex].