Math Problem Statement

Solve the following system of equations graphically: y = -1/2x + 7 and y = 3/2x - 5.

Solution

The system of equations given in the image is:

  1. y=12x+7y = -\frac{1}{2}x + 7
  2. y=32x5y = \frac{3}{2}x - 5

You are tasked to solve this system graphically. To do so, you need to plot both lines on the provided graph and find their intersection point.


Steps to Solve:

  1. Graph the first equation: y=12x+7y = -\frac{1}{2}x + 7

    • Slope: 12-\frac{1}{2} (line decreases as xx increases).
    • y-intercept: 77 (the point (0,7)(0, 7)).
  2. Graph the second equation: y=32x5y = \frac{3}{2}x - 5

    • Slope: 32\frac{3}{2} (line increases as xx increases).
    • y-intercept: 5-5 (the point (0,5)(0, -5)).
  3. Plot both lines:

    • For y=12x+7y = -\frac{1}{2}x + 7, plot the point (0,7)(0, 7) and use the slope 12-\frac{1}{2} to go down 1 unit for every 2 units to the right.
    • For y=32x5y = \frac{3}{2}x - 5, plot the point (0,5)(0, -5) and use the slope 32\frac{3}{2} to go up 3 units for every 2 units to the right.
  4. Find the intersection:

    • The lines intersect where their yy-values are equal, which solves the system of equations. Algebraically, you solve:

    12x+7=32x5-\frac{1}{2}x + 7 = \frac{3}{2}x - 5


Solving for xx:

  1. Add 12x\frac{1}{2}x to both sides:

7=2x57 = 2x - 5

  1. Add 5 to both sides:

12=2x12 = 2x

  1. Divide by 2:

x=6x = 6


Solve for yy:

Substitute x=6x = 6 into either equation, e.g., y=12x+7y = -\frac{1}{2}x + 7:

y=12(6)+7y = -\frac{1}{2}(6) + 7

y=3+7=4y = -3 + 7 = 4


Final Solution:

The intersection point is (6,4)(6, 4).


Do you need help plotting this solution on the graph or further clarification?

Here are 5 related questions for practice:

  1. What does the slope of a line tell you about its steepness and direction?
  2. Solve another system of equations: y=2x+3y = 2x + 3 and y=x1y = -x - 1.
  3. How do you determine the solution of a system graphically if the lines are parallel?
  4. What happens when two equations have the same line (identical)?
  5. Convert 2x3y=62x - 3y = 6 into slope-intercept form.

Tip: When solving graphically, ensure your graph is scaled correctly to make the intersection point clear!

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Graphing Systems of Equations
Slope and Y-intercept

Formulas

Slope-intercept form: y = mx + b
Intersection of lines

Theorems

Two lines intersect where their y-values are equal.

Suitable Grade Level

Grades 8-10