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Answer :
To solve the equation [tex]\(-\frac{1}{2}x + 4 = x + 1\)[/tex], we want to find the value of [tex]\(x\)[/tex] where both equations [tex]\(y = -\frac{1}{2}x + 4\)[/tex] and [tex]\(y = x + 1\)[/tex] intersect.
1. Set the equations equal:
[tex]\[
-\frac{1}{2}x + 4 = x + 1
\][/tex]
2. Rearrange the terms:
Move all terms containing [tex]\(x\)[/tex] to one side of the equation:
[tex]\[
-\frac{1}{2}x - x = 1 - 4
\][/tex]
3. Simplify the equation:
Combine like terms:
[tex]\[
-\frac{3}{2}x = -3
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Multiply both sides by [tex]\(-\frac{2}{3}\)[/tex] to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{-3}{-\frac{3}{2}} = 2
\][/tex]
5. Find the corresponding [tex]\(y\)[/tex]-value:
Substitute [tex]\(x = 2\)[/tex] back into either equation to find [tex]\(y\)[/tex]. Using [tex]\(y = x + 1\)[/tex]:
[tex]\[
y = 2 + 1 = 3
\][/tex]
The solution to the equation [tex]\(-\frac{1}{2}x + 4 = x + 1\)[/tex] is [tex]\((x, y) = (2, 3)\)[/tex]. This means the point of intersection of the graphs of the two equations is at [tex]\((2, 3)\)[/tex].
1. Set the equations equal:
[tex]\[
-\frac{1}{2}x + 4 = x + 1
\][/tex]
2. Rearrange the terms:
Move all terms containing [tex]\(x\)[/tex] to one side of the equation:
[tex]\[
-\frac{1}{2}x - x = 1 - 4
\][/tex]
3. Simplify the equation:
Combine like terms:
[tex]\[
-\frac{3}{2}x = -3
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Multiply both sides by [tex]\(-\frac{2}{3}\)[/tex] to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{-3}{-\frac{3}{2}} = 2
\][/tex]
5. Find the corresponding [tex]\(y\)[/tex]-value:
Substitute [tex]\(x = 2\)[/tex] back into either equation to find [tex]\(y\)[/tex]. Using [tex]\(y = x + 1\)[/tex]:
[tex]\[
y = 2 + 1 = 3
\][/tex]
The solution to the equation [tex]\(-\frac{1}{2}x + 4 = x + 1\)[/tex] is [tex]\((x, y) = (2, 3)\)[/tex]. This means the point of intersection of the graphs of the two equations is at [tex]\((2, 3)\)[/tex].
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