Sure, let's solve the questions step by step.
1. Solution of [tex]\(3x + 5 = 2x - 7\)[/tex]:
To find the solution to this equation, we need to isolate [tex]\(x\)[/tex].
[tex]\[
3x + 5 = 2x - 7
\][/tex]
Subtract [tex]\(2x\)[/tex] from both sides:
[tex]\[
3x - 2x + 5 = -7
\][/tex]
Which simplifies to:
[tex]\[
x + 5 = -7
\][/tex]
Subtract 5 from both sides:
[tex]\[
x = -12
\][/tex]
So, the solution to [tex]\(3x + 5 = 2x - 7\)[/tex] is [tex]\(x = -12\)[/tex].
2. The [tex]\(x\)[/tex]-coordinates of the intersection point for the lines [tex]\(y = 3x + 5\)[/tex] and [tex]\(y = 2x - 7\)[/tex]:
From our solution above, we already found that [tex]\(x = -12\)[/tex]. So the [tex]\(x\)[/tex]-coordinate of the intersection point is:
[tex]\[
x = -12
\][/tex]
3. The [tex]\(x\)[/tex]-coordinates of the [tex]\(x\)[/tex]-intercepts for the lines [tex]\(y = 3x + 5\)[/tex] and [tex]\(y = 2x - 7\)[/tex]:
- For the line [tex]\(y = 3x + 5\)[/tex], to find the [tex]\(x\)[/tex]-intercept, set [tex]\(y = 0\)[/tex]:
[tex]\[
0 = 3x + 5
\][/tex]
[tex]\[
3x = -5
\][/tex]
[tex]\[
x = -\frac{5}{3}
\][/tex]
So, the [tex]\(x\)[/tex]-coordinate of the [tex]\(x\)[/tex]-intercept for [tex]\(y = 3x + 5\)[/tex] is:
[tex]\[
x = -\frac{5}{3} \approx -1.6667
\][/tex]
- For the line [tex]\(y = 2x - 7\)[/tex], to find the [tex]\(x\)[/tex]-intercept, set [tex]\(y = 0\)[/tex]:
[tex]\[
0 = 2x - 7
\][/tex]
[tex]\[
2x = 7
\][/tex]
[tex]\[
x = \frac{7}{2}
\][/tex]
So, the [tex]\(x\)[/tex]-coordinate of the [tex]\(x\)[/tex]-intercept for [tex]\(y = 2x - 7\)[/tex] is:
[tex]\[
x = \frac{7}{2} = 3.5
\][/tex]
4. The [tex]\(y\)[/tex]-coordinate of the intersection point for the lines [tex]\(y = 3x + 5\)[/tex] and [tex]\(y = 2x - 7\)[/tex]:
Using [tex]\(x = -12\)[/tex] in either of the original equations, we get:
- For [tex]\(y = 3x + 5\)[/tex]:
[tex]\[
y = 3(-12) + 5 = -36 + 5 = -31
\][/tex]
- For [tex]\(y = 2x - 7\)[/tex]:
[tex]\[
y = 2(-12) - 7 = -24 - 7 = -31
\][/tex]
So, the [tex]\(y\)[/tex]-coordinate of the intersection point is:
[tex]\[
y = -31
\][/tex]
5. The [tex]\(y\)[/tex]-coordinates of the [tex]\(y\)[/tex]-intercepts for the lines [tex]\(y = 3x + 5\)[/tex] and [tex]\(y = 2x - 7\)[/tex]:
- For the line [tex]\(y = 3x + 5\)[/tex], to find the [tex]\(y\)[/tex]-intercept, set [tex]\(x = 0\)[/tex]:
[tex]\[
y = 3(0) + 5 = 5
\][/tex]
So, the [tex]\(y\)[/tex]-coordinate of the [tex]\(y\)[/tex]-intercept for [tex]\(y = 3x + 5\)[/tex] is:
[tex]\[
y = 5
\][/tex]
- For the line [tex]\(y = 2x - 7\)[/tex], to find the [tex]\(y\)[/tex]-intercept, set [tex]\(x = 0\)[/tex]:
[tex]\[
y = 2(0) - 7 = -7
\][/tex]
So, the [tex]\(y\)[/tex]-coordinate of the [tex]\(y\)[/tex]-intercept for [tex]\(y = 2x - 7\)[/tex] is:
[tex]\[
y = -7
\][/tex]
