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

Question

y, equals, one quarter, x, plus, 4
y=
4
1
​
x+4
3, x, plus, 2, y, equals, minus, 6
3x+2y=−6

Answer 1

To solve the system of equations graphically, we will plot the two equations on a coordinate plane.

1. The first equation is: \[ y = \frac{1}{4}x + 4 \]

2. The second equation is: \[ 3x + 2y = -6 \] We can rewrite this in slope-intercept form (y = mx + b):

First, isolate \(y\): \[ 2y = -3x - 6 \] \[ y = -\frac{3}{2}x - 3 \]

Plotting the Equations:

First Equation: \(y = \frac{1}{4}x + 4\)

• The y-intercept is \(4\) (when \(x=0\)).
• The slope is \(\frac{1}{4}\), which means for every \(4\) units you move right (increasing \(x\)), you move \(1\) unit up in \(y\).

Points to Plot:

• When \(x = 0\), \(y = 4\) → Point: \((0, 4)\)
• When \(x = 4\), \(y = \frac{1}{4}(4) + 4 = 1 + 4 = 5\) → Point: \((4, 5)\)

Second Equation: \(y = -\frac{3}{2}x - 3\)

• The y-intercept is \(-3\) (when \(x=0\)).
• The slope is \(-\frac{3}{2}\), which means for every \(2\) units you move right, you move \(3\) units down in \(y\).

Points to Plot:

• When \(x = 0\), \(y = -3\) → Point: \((0, -3)\)
• When \(x = 2\), \(y = -\frac{3}{2}(2) - 3 = -3 - 3 = -6\) → Point: \((2, -6)\)

Graphing:

1. Plot the points for the first equation \((0, 4)\) and \((4, 5)\) and draw the line.
2. Plot the points for the second equation \((0, -3)\) and \((2, -6)\) and draw the line.
3. The point where the two lines intersect is the solution to the system of equations.

Finding the Intersection:

After plotting the lines, you look for the point of intersection between the two lines. This point is the solution to the system of equations.

Conclusion:

To find the exact coordinates where the two lines intersect, you can either:

• Observe the graph, or
• Solve the equations algebraically.

By substituting values or solving the system, you can determine the exact coordinates of the intersection point visually or mathematically.

If you need further assistance in understanding how to graph these equations or need help with specific steps, feel free to ask!

Answer 2

whats the number where they intersect

Answer 3

To find the intersection point of the two equations algebraically, we have:

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

2. The second equation reformulated: \[ y = -\frac{3}{2}x - 3 \]

Since both expressions are equal to \(y\), we can set them equal to each other to find \(x\):

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

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

1. To eliminate the fractions, multiply the entire equation by \(4\) (the least common multiple): \[ x + 16 = -6x - 12 \]

2. Combine like terms: \[ x + 6x = -12 - 16 \] \[ 7x = -28 \]

3. Divide both sides by \(7\): \[ x = -4 \]

Now that we have \(x = -4\), substitute this value back into either of the original equations to find \(y\). We'll use the first equation:

\[ y = \frac{1}{4}(-4) + 4 \] \[ y = -1 + 4 = 3 \]

Intersection Point

Thus, the intersection point of the two lines is: \[ \boxed{(-4, 3)} \]