Answer :
To solve the equation [tex]\(-\frac{1}{2}x + 4 = x + 1\)[/tex] by finding the intersection of the two lines [tex]\(y = -\frac{1}{2}x + 4\)[/tex] and [tex]\(y = x + 1\)[/tex], we can follow these steps:
1. Set the equations equal to each other: Since both equations are equal to [tex]\(y\)[/tex], set the right-hand sides equal:
[tex]\[
-\frac{1}{2}x + 4 = x + 1
\][/tex]
2. Rearrange and combine like terms: Move all terms involving [tex]\(x\)[/tex] to one side and constant terms to the other side:
[tex]\[
-\frac{1}{2}x - x = 1 - 4
\][/tex]
Simplify the equation:
[tex]\[
-\frac{1}{2}x - x = -3
\][/tex]
3. Combine the [tex]\(x\)[/tex] terms: Combine the coefficients of [tex]\(x\)[/tex]:
[tex]\[
-\frac{3}{2}x = -3
\][/tex]
4. Solve for [tex]\(x\)[/tex]: Divide both sides by [tex]\(-\frac{3}{2}\)[/tex] to isolate [tex]\(x\)[/tex]:
[tex]\[
x = \frac{-3}{-\frac{3}{2}}
\][/tex]
This simplifies to:
[tex]\[
x = 2
\][/tex]
5. Find the [tex]\(y\)[/tex] coordinate: Substitute [tex]\(x = 2\)[/tex] back into either of the original equations to find [tex]\(y\)[/tex]. We'll use [tex]\(y = x + 1\)[/tex]:
[tex]\[
y = 2 + 1 = 3
\][/tex]
6. Write the solution: The solution for the system, and thus the original equation, is the point where the two lines intersect: [tex]\((x, y) = (2, 3)\)[/tex].
So, the solution of the equation [tex]\(-\frac{1}{2}x + 4 = x + 1\)[/tex] is [tex]\((2, 3)\)[/tex].
1. Set the equations equal to each other: Since both equations are equal to [tex]\(y\)[/tex], set the right-hand sides equal:
[tex]\[
-\frac{1}{2}x + 4 = x + 1
\][/tex]
2. Rearrange and combine like terms: Move all terms involving [tex]\(x\)[/tex] to one side and constant terms to the other side:
[tex]\[
-\frac{1}{2}x - x = 1 - 4
\][/tex]
Simplify the equation:
[tex]\[
-\frac{1}{2}x - x = -3
\][/tex]
3. Combine the [tex]\(x\)[/tex] terms: Combine the coefficients of [tex]\(x\)[/tex]:
[tex]\[
-\frac{3}{2}x = -3
\][/tex]
4. Solve for [tex]\(x\)[/tex]: Divide both sides by [tex]\(-\frac{3}{2}\)[/tex] to isolate [tex]\(x\)[/tex]:
[tex]\[
x = \frac{-3}{-\frac{3}{2}}
\][/tex]
This simplifies to:
[tex]\[
x = 2
\][/tex]
5. Find the [tex]\(y\)[/tex] coordinate: Substitute [tex]\(x = 2\)[/tex] back into either of the original equations to find [tex]\(y\)[/tex]. We'll use [tex]\(y = x + 1\)[/tex]:
[tex]\[
y = 2 + 1 = 3
\][/tex]
6. Write the solution: The solution for the system, and thus the original equation, is the point where the two lines intersect: [tex]\((x, y) = (2, 3)\)[/tex].
So, the solution of the equation [tex]\(-\frac{1}{2}x + 4 = x + 1\)[/tex] is [tex]\((2, 3)\)[/tex].
