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What is the solution of [tex]\(3x + 5 = 2x - 7\)[/tex]?

A. The [tex]\(x\)[/tex]-coordinates of the intersection point of the lines [tex]\(y = 3x + 5\)[/tex] and [tex]\(y = 2x - 7\)[/tex]

B. 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]

C. The [tex]\(y\)[/tex]-coordinate of the intersection point of the lines [tex]\(y = 3x + 5\)[/tex] and [tex]\(y = 2x - 7\)[/tex]

D. 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]


Sagot :

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]