High School

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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 :

Sure! Let's solve the equation [tex]\( -\frac{1}{2} x + 4 = x + 1 \)[/tex] step-by-step.

1. Start with the given equation:

[tex]\[
-\frac{1}{2} x + 4 = x + 1
\][/tex]

2. To eliminate the fraction, multiply every term by 2:

[tex]\[
2 \left( -\frac{1}{2} x \right) + 2 \cdot 4 = 2 \cdot x + 2 \cdot 1
\][/tex]

This simplifies to:

[tex]\[
-x + 8 = 2x + 2
\][/tex]

3. Next, we'll move all the [tex]\(x\)[/tex]-terms to one side and constant terms to the other side. Add [tex]\(x\)[/tex] to both sides:

[tex]\[
-x + x + 8 = 2x + x + 2
\][/tex]

Simplifying gives:

[tex]\[
8 = 3x + 2
\][/tex]

4. Subtract 2 from both sides to isolate the term with [tex]\(x\)[/tex]:

[tex]\[
8 - 2 = 3x + 2 - 2
\][/tex]

Which simplifies to:

[tex]\[
6 = 3x
\][/tex]

5. Finally, divide both sides by 3 to solve for [tex]\(x\)[/tex]:

[tex]\[
\frac{6}{3} = \frac{3x}{3}
\][/tex]

This gives:

[tex]\[
x = 2
\][/tex]

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

[tex]\[
x = 2
\][/tex]

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