Let's break down and solve each component step-by-step:
1. Finding the solution to the equation [tex]\(3x + 5 = 2x - 7\)[/tex]:
To find the solution for [tex]\( x \)[/tex]:
[tex]\[
3x + 5 = 2x - 7
\][/tex]
Subtract [tex]\( 2x \)[/tex] from both sides:
[tex]\[
x + 5 = -7
\][/tex]
Subtract 5 from both sides:
[tex]\[
x = -12
\][/tex]
The solution for [tex]\( x \)[/tex] is [tex]\(-12\)[/tex].
2. Finding the [tex]\( x \)[/tex]-coordinates of the intersection point of the lines [tex]\( y = 3x + 5 \)[/tex] and [tex]\( y = 2x - 7 \)[/tex]:
The lines intersect where their [tex]\( y \)[/tex] values are the same. Since we have already solved for [tex]\( x \)[/tex], the [tex]\( x \)[/tex]-coordinate of the intersection point is the same [tex]\( x \)[/tex] we found:
[tex]\[
x = -12
\][/tex]
3. Finding the [tex]\( x \)[/tex]-coordinates of the [tex]\( x \)[/tex]-intercepts of the lines [tex]\( y = 3x + 5 \)[/tex] and [tex]\( y = 2x - 7 \)[/tex]:
The [tex]\( x \)[/tex]-intercept is the point where [tex]\( y = 0 \)[/tex].
- For [tex]\( y = 3x + 5 \)[/tex]:
[tex]\[
0 = 3x + 5 \quad \Rightarrow \quad 3x = -5 \quad \Rightarrow \quad x = \frac{-5}{3} \approx -1.67
\][/tex]
- For [tex]\( y = 2x - 7 \)[/tex]:
[tex]\[
0 = 2x - 7 \quad \Rightarrow \quad 2x = 7 \quad \Rightarrow \quad x = \frac{7}{2} = 3.5
\][/tex]
4. Finding the [tex]\( y \)[/tex]-coordinate of the intersection point of the lines [tex]\( y = 3x + 5 \)[/tex] and [tex]\( y = 2x - 7 \)[/tex]:
Using the [tex]\( x \)[/tex]-coordinate [tex]\( x = -12 \)[/tex] in either of the line equations (let's use [tex]\( y = 3x + 5 \)[/tex]):
[tex]\[
y = 3(-12) + 5 = -36 + 5 = -31
\][/tex]
The [tex]\( y \)[/tex]-coordinate of the intersection point is [tex]\(-31\)[/tex].
5. Finding the [tex]\( y \)[/tex]-coordinates of the [tex]\( y \)[/tex]-intercepts of the lines [tex]\( y = 3x + 5 \)[/tex] and [tex]\( y = 2x - 7 \)[/tex]:
The [tex]\( y \)[/tex]-intercept is the constant term in the line equations when [tex]\( x = 0 \)[/tex].
- For [tex]\( y = 3x + 5 \)[/tex]:
[tex]\[
y = 5
\][/tex]
- For [tex]\( y = 2x - 7 \)[/tex]:
[tex]\[
y = -7
\][/tex]
### Summary of Answers:
1. Solution [tex]\( x \)[/tex] of [tex]\( 3x + 5 = 2x - 7 \)[/tex]:
[tex]\[
x = -12
\][/tex]
2. [tex]\( x \)[/tex]-coordinate of the intersection point:
[tex]\[
x = -12
\][/tex]
3. [tex]\( x \)[/tex]-coordinates of the [tex]\( x \)[/tex]-intercepts:
- For [tex]\( y = 3x + 5 \)[/tex]:
[tex]\[
x \approx -1.67
\][/tex]
- For [tex]\( y = 2x - 7 \)[/tex]:
[tex]\[
x = 3.5
\][/tex]
4. [tex]\( y \)[/tex]-coordinate of the intersection point:
[tex]\[
y = -31
\][/tex]
5. [tex]\( y \)[/tex]-coordinates of the [tex]\( y \)[/tex]-intercepts:
- For [tex]\( y = 3x + 5 \)[/tex]:
[tex]\[
y = 5
\][/tex]
- For [tex]\( y = 2x - 7 \)[/tex]:
[tex]\[
y = -7
\][/tex]
