Question

he system of linear equations 3x + 2y = −6 and y equals one half times x plus 4 is graphed on a coordinate plane. Approximate the solution to the system.

coordinate plane with one line that passes through the points 0 comma 4 and 2 comma 5 and another line that passes through the points 0 comma negative 3 and negative 2 comma 0

Answer 1

To find the approximate solution to the system of linear equations given by

1. \(3x + 2y = -6\)
2. \(y = \frac{1}{2}x + 4\)

we can solve the first equation for \(y\) in order to compare both lines more easily.

1. **From the first equation \(3x + 2y = -6\):**

Rearranging gives:
\[
2y = -3x - 6
\]
\[
y = -\frac{3}{2}x - 3
\]

The slope of this line is \(-\frac{3}{2}\) and the y-intercept is \(-3\).

2. **The second equation is already in slope-intercept form:**

The line \(y = \frac{1}{2}x + 4\) has a slope of \(\frac{1}{2}\) and a y-intercept of \(4\).

Now we can identify and approximate the intersection point of the two lines. From your description:
- The \(y = \frac{1}{2}x + 4\) line passes through points \((0, 4)\) and \((2, 5)\).
- The line from the first equation (rewritten) passes through points \((0, -3)\) and \((-2, 0)\).

### Graphical Approximation
Based on your coordinate points, we can graph both lines:

1. The first line ( \(3x + 2y = -6\) or \(y = -\frac{3}{2}x - 3\) ):
- At \(x=0\), \(y = -3\).
- At \(x=-2\), \(y=0\).

2. The second line (\(y = \frac{1}{2}x + 4\)):
- At \(x=0\), \(y = 4\).
- At \(x=2\), \(y = 5\).

### Finding the Approximate Intersection
To find the approximate intersection, substitute values or look for where the two lines seem to cross:

- We can set both equations equal to each other to find an exact solution for further accuracy.

\[
-\frac{3}{2}x - 3 = \frac{1}{2}x + 4
\]

Combining terms:
\[
-\frac{3}{2}x - \frac{1}{2}x = 4 + 3
\]
\[
-\frac{4}{2}x = 7
\]
\[
-2x = 7
\]
\[
x = -\frac{7}{2} = -3.5
\]

Substituting \(x = -3.5\) back into one of the equations to find \(y\):

\[
y = \frac{1}{2}(-3.5) + 4 = -1.75 + 4 = 2.25
\]

Therefore, the approximate solution to the system of equations is:

\[
\boxed{\left(-3.5, 2.25\right)}
\]

This means the two lines intersect at approximately \((-3.5, 2.25)\) on the coordinate plane.