How do you graph the system of equations y = 2x + 3 and 2x + 4y = 8?

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To graph the system of equations given by y = 2x + 3 and 2x + 4y = 8, we need to first understand the individual graphs of these equations.

Step 1: Understand the First Equation

The first equation, y = 2x + 3, is already in slope-intercept form, where:

  • slope (m) = 2
  • y-intercept (b) = 3

This means that the graph of this line will rise 2 units up for every 1 unit it moves to the right. To plot it:

  1. Start at the y-intercept (0, 3) on the y-axis.
  2. From this point, use the slope to find another point: move 1 unit to the right (to x = 1), and then 2 units up (to y = 5). Therefore, the second point is (1, 5).
  3. Draw a line through these points, extending it in both directions.

Step 2: Transform the Second Equation

The second equation, 2x + 4y = 8, needs to be rewritten in slope-intercept form:

  1. Subtract 2x from both sides:

    2. 4y = -2x + 8

  2. Divide all terms by 4:

    3. y = -0.5x + 2

Step 3: Analyze the Second Equation

This equation is also in slope-intercept form with:

  • slope (m) = -0.5
  • y-intercept (b) = 2

To plot this line:

  1. Start at the y-intercept (0, 2).
  2. From this point, use the slope: move 1 unit to the right (to x = 1), and then move down 0.5 units (to y = 1.5). So, another point is (1, 1.5).
  3. Draw a line through these points.

Step 4: Plot the Graphs

Now that we have the equations reformed:

  • The line for y = 2x + 3 will be steep and rising.
  • The line for y = -0.5x + 2 will be more gradual and falling.

Step 5: Find the Intersection

To find the solution to the system (the point where these lines intersect), set the equations equal to each other:

2x + 3 = -0.5x + 2

Now solve for x:

  • Add 0.5x to both sides: 2.5x + 3 = 2
  • Subtract 3 from both sides: 2.5x = -1
  • Divide by 2.5: x = -0.4

Substitute this value back into one of the original equations (let’s use the first one):

y = 2(-0.4) + 3 = -0.8 + 3 = 2.2

So, the point of intersection is (-0.4, 2.2).

Final Graph

When you graph both equations, you should see:

  • The line y = 2x + 3 rising steeply.
  • The line y = -0.5x + 2 descending gradually.
  • Both lines intersect at the point (-0.4, 2.2) which is the solution to the system.

Make sure to label your axes and the intersection point for clarity!

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