Math Problem Statement

Find two pairs of coordinates for each equation by making a T-chart. Use the coordinates to graph the lines and find the solution. x + 2y = 6 2x - 3y = 12

Solution

To solve the system of equations by graphing, we'll first create T-charts for each equation, find pairs of coordinates, and then use these to graph the lines.

Step 1: Solve for y in each equation.

  1. First equation: x+2y=6x + 2y = 6

    Solve for yy:

    2y=6x2y = 6 - x y=6x2y = \frac{6 - x}{2}

  2. Second equation: 2x3y=122x - 3y = 12

    Solve for yy:

    3y=122x-3y = 12 - 2x y=2x123y = \frac{2x - 12}{3}

Step 2: Create T-Charts and Find Coordinates

  1. For x+2y=6x + 2y = 6:

    Choose values for xx and find corresponding yy values.

    xxyyCoordinates
    03(0, 3)
    22(2, 2)
    41(4, 1)
  2. For 2x3y=122x - 3y = 12:

    Choose values for xx and find corresponding yy values.

    xxyyCoordinates
    0-4(0, -4)
    3-2(3, -2)
    60(6, 0)

Step 3: Graph the Lines and Find the Solution

  • Plot the coordinates from both T-charts on a graph.
  • Draw lines through the points for each equation.
  • The intersection point of these two lines will be the solution to the system.

If you graph these correctly, you will find that the lines intersect at the point (6,0)(6, 0), which is the solution to the system of equations.

Solution:

The solution is (6,0)(6, 0).


Do you have any questions, or would you like further details on any part of this process?

Related Questions:

  1. What happens if two lines do not intersect?
  2. How can we verify the solution using substitution?
  3. Can the solution be a fraction or decimal?
  4. What if the lines overlap completely?
  5. What method can be used if graphing is not practical?

Tip: Always double-check your graphs by substituting the intersection point back into both original equations!

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Math Problem Analysis

Mathematical Concepts

Linear equations
Graphing lines

Formulas

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Theorems

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Suitable Grade Level

Grades 7-9