Math Problem Statement

For the following two lines, find the point where they intersect and graph: y1 = 2 - x and y2 = 3x - 2.

Solution

The problem in your image involves finding the point where two lines intersect, with equations given as:

y1=2xandy2=3x2y_1 = 2 - x \quad \text{and} \quad y_2 = 3x - 2

Step 1: Set the equations equal to each other.

To find the intersection, we set y1=y2y_1 = y_2 because at the point of intersection, both yy-values are the same:

2x=3x22 - x = 3x - 2

Step 2: Solve for xx.

Rearrange the terms to solve for xx:

2x=3x2    2+2=3x+x    4=4x    x=12 - x = 3x - 2 \implies 2 + 2 = 3x + x \implies 4 = 4x \implies x = 1

Step 3: Substitute x=1x = 1 into either equation to find yy.

Substitute x=1x = 1 into either y1=2xy_1 = 2 - x or y2=3x2y_2 = 3x - 2:

  • Using y1=2xy_1 = 2 - x:

y1=21=1y_1 = 2 - 1 = 1

Thus, the point of intersection is (1,1)(1, 1).

Step 4: Graph the lines.

Now, we will graph the two lines using their equations:

  • For y1=2xy_1 = 2 - x, this is a line with slope 1-1 and y-intercept 22.
  • For y2=3x2y_2 = 3x - 2, this is a line with slope 33 and y-intercept 2-2.

Both lines will intersect at the point (1,1)(1, 1).

Would you like more details or help with graphing these lines?

Follow-up Questions:

  1. Can you solve for the intersection of any two lines by setting the equations equal to each other?
  2. What is the geometric significance of the slopes of the two lines in this problem?
  3. How does changing the slope or intercept affect the intersection point?
  4. Can you verify the solution graphically by plotting both lines on the grid?
  5. How would you find the intersection if the lines were parallel or identical?

Tip: The intersection point of two lines represents the solution to the system of equations, as the point satisfies both equations simultaneously.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Systems of Equations
Graphing

Formulas

y1 = 2 - x
y2 = 3x - 2

Theorems

Intersection of two lines
Solving systems by substitution

Suitable Grade Level

Grades 7-10