Answer :
To solve the equation [tex]\(-\frac{1}{2} x + 4 = x + 1\)[/tex], we can use the method of graphing the equations separately and then finding their intersection point.
1. Understand the Equations:
- The first equation is [tex]\(y = -\frac{1}{2} x + 4\)[/tex]. This is a linear equation with a slope of [tex]\(-\frac{1}{2}\)[/tex] and a y-intercept of 4.
- The second equation is [tex]\(y = x + 1\)[/tex]. This one also is linear, with a slope of 1 and a y-intercept of 1.
2. Graph the Equations:
- Plot the first line, [tex]\(y = -\frac{1}{2} x + 4\)[/tex]. It will slope downwards from left to right, starting at the point [tex]\((0, 4)\)[/tex] where it crosses the y-axis.
- Plot the second line, [tex]\(y = x + 1\)[/tex]. This line will slope upwards, starting at [tex]\((0, 1)\)[/tex], its y-intercept.
3. Find the Intersection:
- The solution to the system of equations is the point where these two lines intersect.
- By observation or calculation, the intersection occurs at the point [tex]\((2, 3)\)[/tex].
4. Verify the Solution:
- Substitute [tex]\(x = 2\)[/tex] back into the original equation to confirm that both sides are equal:
[tex]\[
-\frac{1}{2} \times 2 + 4 = 2 + 1
\][/tex]
Simplifying both sides:
[tex]\[
-1 + 4 = 3 \quad \text{and} \quad 2 + 1 = 3
\][/tex]
Both sides equal 3, confirming that the solution is correct.
Therefore, the solution to the equation [tex]\(-\frac{1}{2} x + 4 = x + 1\)[/tex] is [tex]\(x = 2\)[/tex].
1. Understand the Equations:
- The first equation is [tex]\(y = -\frac{1}{2} x + 4\)[/tex]. This is a linear equation with a slope of [tex]\(-\frac{1}{2}\)[/tex] and a y-intercept of 4.
- The second equation is [tex]\(y = x + 1\)[/tex]. This one also is linear, with a slope of 1 and a y-intercept of 1.
2. Graph the Equations:
- Plot the first line, [tex]\(y = -\frac{1}{2} x + 4\)[/tex]. It will slope downwards from left to right, starting at the point [tex]\((0, 4)\)[/tex] where it crosses the y-axis.
- Plot the second line, [tex]\(y = x + 1\)[/tex]. This line will slope upwards, starting at [tex]\((0, 1)\)[/tex], its y-intercept.
3. Find the Intersection:
- The solution to the system of equations is the point where these two lines intersect.
- By observation or calculation, the intersection occurs at the point [tex]\((2, 3)\)[/tex].
4. Verify the Solution:
- Substitute [tex]\(x = 2\)[/tex] back into the original equation to confirm that both sides are equal:
[tex]\[
-\frac{1}{2} \times 2 + 4 = 2 + 1
\][/tex]
Simplifying both sides:
[tex]\[
-1 + 4 = 3 \quad \text{and} \quad 2 + 1 = 3
\][/tex]
Both sides equal 3, confirming that the solution is correct.
Therefore, the solution to the equation [tex]\(-\frac{1}{2} x + 4 = x + 1\)[/tex] is [tex]\(x = 2\)[/tex].
