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

y equals negative one fourth times x plus 1
y = −2x − 1
a coordinate grid with one line that passes through the points 0 comma 1 and 4 comma 0 and another line that passes through the points 0 comma negative 1 and 1 comma negative 3
a coordinate grid with one line that passes through the points 0 comma 1 and 3 comma 0 and another line that passes through the points 0 comma negative 3 and 1 comma negative 5
a coordinate grid with one line that passes through the points 0 comma 1 and 3 comma negative 1 and another line that passes through the points 0 comma negative 1 and 2 comma negative 5
a coordinate grid with one line that passes through the points 0 comma 1 and 4 comma negative 2 and another line that passes through the points 0 comma negative 2 and 1 comma negative 5

1 answer

To determine which graph correctly represents the solution to the given system of linear equations, we will analyze both equations separately.

  1. For the first equation: \[ y = -\frac{1}{4}x + 1 \]

    • When \( x = 0 \), \( y = 1 \) (point \((0, 1)\)).
    • When \( x = 4 \), \( y = 0 \) (point \((4, 0)\)).

  2. For the second equation: \[ y = -2x - 1 \]

    • When \( x = 0 \), \( y = -1 \) (point \((0, -1)\)).
    • When \( x = 1 \), \( y = -3 \) (point \((1, -3)\)).

Now let's match these points with the options provided:

  • The coordinates found for the first line are \((0, 1)\) and \((4, 0)\).
  • The coordinates found for the second line are \((0, -1)\) and \((1, -3)\).

Next, let’s check each option against these derived points:

  1. First option:

    • Line 1: \((0, 1)\) and \((4, 0)\) → Matches the first line.
    • Line 2: \((0, -1)\) and \((1, -3)\) → Matches the second line.
    • This option is correct.

  2. Second option:

    • Line 1: \((0, 1)\) and \((3, 0)\) → The second point does not match.
    • This option is incorrect.

  3. Third option:

    • Line 1: \((0, 1)\) and \((3, -1)\) → The second point does not match.
    • This option is incorrect.

  4. Fourth option:

    • Line 1: \((0, 1)\) and \((4, -2)\) → The second point does not match.
    • This option is incorrect.

After reviewing all the options, the first option is the only one that accurately represents the system of equations given. Thus, the correct answer is the first graph.