Math Problem Statement

Deshawn is using a coordinate plane to design a mural. Two of the straight lines on the mural are represented by the equations y = 2x + 8 and y = (1/2)x - 4. Graph the equations on the grid so that the point of intersection of the lines is shown.

Solution

The problem involves graphing two linear equations y=2x+8y = 2x + 8 and y=12x4y = \frac{1}{2}x - 4 on a coordinate plane and determining their point of intersection. Let me guide you step by step:


Step 1: Equations to graph

  1. Equation 1: y=2x+8y = 2x + 8

    • This is a line with a slope of 2 and a y-intercept of 8. Start by plotting the point (0, 8), then use the slope riserun=2=21\frac{\text{rise}}{\text{run}} = 2 = \frac{2}{1} to find additional points, such as:
      • From (0, 8), go up 2 units and right 1 unit to reach (1, 10).
      • Alternatively, go down 2 units and left 1 unit to reach (-1, 6).
  2. Equation 2: y=12x4y = \frac{1}{2}x - 4

    • This is a line with a slope of 12\frac{1}{2} and a y-intercept of -4. Start by plotting the point (0, -4), then use the slope riserun=12\frac{\text{rise}}{\text{run}} = \frac{1}{2}:
      • From (0, -4), go up 1 unit and right 2 units to reach (2, -3).
      • Alternatively, go down 1 unit and left 2 units to reach (-2, -5).

Step 2: Intersection point

To find the intersection, solve the equations y=2x+8y = 2x + 8 and y=12x4y = \frac{1}{2}x - 4 algebraically:

  1. Set the equations equal: 2x+8=12x42x + 8 = \frac{1}{2}x - 4

  2. Eliminate the fraction by multiplying the entire equation by 2: 4x+16=x84x + 16 = x - 8

  3. Simplify and solve for xx: 4xx=8163x=24x=84x - x = -8 - 16 \quad \Rightarrow \quad 3x = -24 \quad \Rightarrow \quad x = -8

  4. Substitute x=8x = -8 into one of the equations (e.g., y=2x+8y = 2x + 8) to find yy: y=2(8)+8y=16+8y=8y = 2(-8) + 8 \quad \Rightarrow \quad y = -16 + 8 \quad \Rightarrow \quad y = -8

Thus, the point of intersection is (8,8)(-8, -8).


Step 3: Graph the lines

  • Plot the lines for y=2x+8y = 2x + 8 and y=12x4y = \frac{1}{2}x - 4.
  • Verify they intersect at (8,8)(-8, -8).

Would you like further clarification or assistance with graphing? Let me know!


Follow-Up Questions:

  1. What is the significance of the slopes of the two lines?
  2. How do you confirm the intersection point visually from the graph?
  3. Can two parallel lines have an intersection point? Why or why not?
  4. What happens if both equations are the same (coincident lines)?
  5. How would you graph a vertical line (e.g., x=cx = c)?

Tip: Always double-check your graph to ensure it reflects the calculated intersection point.

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Math Problem Analysis

Mathematical Concepts

Algebra
Graphing Linear Equations
Finding Intersection Points

Formulas

Slope-intercept form y = mx + b
Solving linear equations algebraically

Theorems

Linear Equation Theory

Suitable Grade Level

Grades 8-10