High School

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 :

We start with the equation

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

Step 1. Clear the Fraction

Multiply both sides by 2 to eliminate the fraction:

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

This simplifies to

[tex]$$
-x + 8 = 2x + 2.
$$[/tex]

Step 2. Collect the [tex]\(x\)[/tex]-terms

Subtract [tex]\(2x\)[/tex] from both sides so that all terms containing [tex]\(x\)[/tex] are on one side:

[tex]$$
-x - 2x + 8 = 2.
$$[/tex]

This gives

[tex]$$
-3x + 8 = 2.
$$[/tex]

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

Subtract 8 from both sides to isolate the term with [tex]\(x\)[/tex]:

[tex]$$
-3x = 2 - 8,
$$[/tex]

which simplifies to

[tex]$$
-3x = -6.
$$[/tex]

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

Divide both sides by [tex]\(-3\)[/tex]:

[tex]$$
x = \frac{-6}{-3} = 2.
$$[/tex]

Thus, the solution to the equation

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

is

[tex]$$
x = 2.
$$[/tex]

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