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Solve the following system of equations graphically on the set of axes below.

y, equals, one half, x, minus, 5
y=
2
1
​
x−5
y, equals, minus, x, plus, 4
y=−x+4
Plot two lines by clicking the graph.
Click a line to delete it.
give detailed instructions on how to graph it

1 answer

To solve the system of equations graphically, you can follow these detailed instructions to plot the lines on a set of axes.

Equations to Graph:

  1. \( y = \frac{1}{2} x - 5 \)
  2. \( y = -x + 4 \)

Step-by-Step Instructions:

Step 1: Set Up the Axes

  • Draw a Cartesian coordinate system with a horizontal axis (x-axis) and a vertical axis (y-axis).
  • Label the axes appropriately.
  • Make sure to include a scale on both axes that is suitable for the range of the lines you'll be plotting. For example, you might use increments of 1 for both axes, but depending on your graphing area, you might want larger increments (e.g., 2 or 5).

Step 2: Graph the First Equation \( y = \frac{1}{2}x - 5 \)

  • Identify the y-intercept: The y-intercept is the point where the line crosses the y-axis. For this equation, the y-intercept is \(-5\). Plot the point (0, -5).
  • Find another point: Choose a value for \( x \) to find a corresponding \( y \) value. For example, let \( x = 2 \):
    • \( y = \frac{1}{2}(2) - 5 = 1 - 5 = -4 \)
    • Plot the point (2, -4).
  • Draw the line: Draw a straight line through the points (0, -5) and (2, -4), extending it in both directions.

Step 3: Graph the Second Equation \( y = -x + 4 \)

  • Identify the y-intercept: The y-intercept is \( 4 \). Plot the point (0, 4).
  • Find another point: Choose a value for \( x \) to find a corresponding \( y \) value. For example, let \( x = 2 \):
    • \( y = -2 + 4 = 2 \)
    • Plot the point (2, 2).
  • Draw the line: Draw a straight line through the points (0, 4) and (2, 2), extending it in both directions.

Step 4: Identify the Intersection Point

  • The point where the two lines cross represents the solution to the system of equations.
  • Look for the coordinates of the intersection point by observing where the two lines meet on the graph.

Final Notes

  • Make sure to label each line with its equation for clarity.
  • If you're doing this on graph paper or a digital tool, ensure that your lines are straight, and your points are accurately placed to get a precise representation of the solution.
  • The intersection point gives you the solution to the system of equations, which is the values of \( x \) and \( y \) that satisfy both equations.

If you have a specific graphing tool you're using, these general instructions can still apply, but the steps for plotting or drawing may be slightly different based on the tool's interface.