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Which graph shows the solution to the system of linear equations?



y equals one half times x

y equals negative one half times x minus 4
coordinate plane with one line that passes through the points 0 comma negative 4 and 2 comma negative 5 and another line that passes through the points 0 comma 0 and 2 comma 1
coordinate plane with one line that passes through the points 0 comma 2 and negative 3 comma 3 and another line that passes through the points 0 comma 0 and negative 3 comma negative 1
coordinate plane with one line that passes through the points 3 comma negative 3 and 0 comma negative 2 and another line that passes through the points 0 comma 0 and 3 comma 1
coordinate plane with one line that passes through the points 0 comma 4 and negative 1 comma 1 and another line that passes through the points 0 comma 0 and 1 comma 3

1 answer

To determine which graph shows the solution to the given system of linear equations, we first need to analyze the two equations:

  1. Equation 1: \( y = \frac{1}{2}x \)
    This line passes through the origin (0, 0) with a slope of \( \frac{1}{2} \). Therefore, some points on this line include:

    • (0, 0)
    • (2, 1)
    • (4, 2)
    • etc.

  2. Equation 2: \( y = -\frac{1}{2}x - 4 \)
    This line has a y-intercept of -4 and a slope of \( -\frac{1}{2} \). It passes through:

    • (0, -4) (y-intercept)
    • (2, -5) (when x=2, y = -1 - 4 = -5)
    • (-2, -3) (when x=-2, y = 1 - 4 = -3)
    • etc.

Now, we will look at the options provided to find a graph that contains:

  • The first line passing through (0, 0) and (2, 1).
  • The second line passing through (0, -4) and (2, -5).

From the descriptions:

  1. Option 1: Shows a line passing through (0, -4) and (2, -5) as well as a line that passes through (0, 0) and (2, 1). This matches both equations.

  2. Option 2: Contains a line passing through (0, 0) and (-3, -1), which does not correspond to either equation.

  3. Option 3: Has a line passing through (0, 0) and (3, 1), which does not correspond to the equations.

  4. Option 4: Has a line through (0, 4) and (-1, 1), which does not correspond to the equations.

Thus, Option 1 is the correct graph representing the system of linear equations.