Ask a New Question
Menu

What is the apparent solution to the system of equations?

y=12x+2y=2x−1

Graph the system of equations using the Line tool. Plot a point at the apparent solution to the system using the Point tool.

Keyboard Instructions
Initial graph state
The horizontal axis goes from -10.8 to 10.8 with ticks spaced every 1 unit(s).
The vertical axis goes from -10.8 to 10.8 with ticks spaced every 1 unit(s).

1 answer

To solve the system of equations given by:

  1. \(y = 12x + 2\)
  2. \(y = 2x - 1\)

we will set the two equations equal to each other to find the intersection point, which is the solution to the system:

\[ 12x + 2 = 2x - 1 \]

Now, we'll solve for \(x\):

  1. Move \(2x\) to the left side: \[ 12x - 2x + 2 = -1 \] which simplifies to: \[ 10x + 2 = -1 \]

  2. Next, subtract \(2\) from both sides: \[ 10x = -3 \]

  3. Now, divide by \(10\): \[ x = -\frac{3}{10} = -0.3 \]

Now that we have the \(x\)-coordinate, we can substitute \(x = -0.3\) back into either of the original equations to find the \(y\)-coordinate. We'll use the second equation:

\[ y = 2(-0.3) - 1 = -0.6 - 1 = -1.6 \]

So, the apparent solution to the system of equations is: \[ (x, y) = \left(-0.3, -1.6\right) \]

Graphing the System of Equations

  1. Graph the First Equation \(y = 12x + 2\):

    • When \(x = 0\): \(y = 2\)
    • When \(x = 1\): \(y = 14\)
    • When \(x = -1\): \(y = -10\)

  2. Graph the Second Equation \(y = 2x - 1\):

    • When \(x = 0\): \(y = -1\)
    • When \(x = 1\): \(y = 1\)
    • When \(x = -1\): \(y = -3\)

Plotting Points

After plotting both lines based on the coordinates calculated:

  • The line for \(y = 12x + 2\) is steep, and crosses the y-axis at (0,2).
  • The line for \(y = 2x - 1\) rises more gradually and crosses the y-axis at (0,-1).

To locate the apparent solution on the graph:

  • Plot the point \((-0.3, -1.6)\).

This point lies somewhere between the intersections of the two lines, where they intersect at the calculated coordinates.

Final Step

Ensure to highlight the point \((-0.3, -1.6)\) clearly while graphing. If you're using digital drawing or graphing software, you can enable the line and point tools to accomplish this.