To solve the system of equations graphically, we will first write both equations in slope-intercept form (y = mx + b).
The system of equations is:
- \( 4x - y = -4 \)
- \( 2x - 3y = -4 \)
Step 1: Rearranging the Equations
Equation 1: \[ 4x - y = -4 \] To express \(y\) in terms of \(x\), rewrite it as: \[ y = 4x + 4 \]
Equation 2: \[ 2x - 3y = -4 \] Now, solve for \(y\): \[ -3y = -2x - 4 \ y = \frac{2}{3}x + \frac{4}{3} \]
Step 2: Graphing the Equations
Graphing Equation 1: The equation \(y = 4x + 4\) has a slope (m) of 4 and a y-intercept of 4. Start at the point (0, 4) and rise 4 units up for every 1 unit you move to the right.
Graphing Equation 2: The equation \(y = \frac{2}{3}x + \frac{4}{3}\) has a slope of \(\frac{2}{3}\) and a y-intercept of \(\frac{4}{3}\), which is approximately \(1.33\). Start at (0, \(\frac{4}{3}\)) and for every 3 units you move to the right you rise 2 units up.
Step 3: Finding the Intersection Point
To find the solution graphically, plot both lines on a graph. The point where these two lines intersect is the solution to the system of equations.
Example of How to Graph:
- Draw the axes (x and y).
- Plot the y-intercept of the first equation (0, 4) and use the slope to plot another point.
- Plot the y-intercept of the second equation (0, \( \frac{4}{3} \)) and use its slope to plot another point.
- Draw straight lines through the points for each equation.
- Identify the point where the two lines intersect.
Additional Calculation:
If you want to find the intersection point algebraically, you can solve the system:
From the first equation: \( y = 4x + 4 \)
Substituting into the second equation: \[ 2x - 3(4x + 4) = -4 \ 2x - 12x - 12 = -4 \ -10x - 12 = -4 \ -10x = 8 \ x = -\frac{4}{5} \]
Now substitute \(x\) back into one of the original equations to find \(y\): \[ y = 4(-\frac{4}{5}) + 4 = -\frac{16}{5} + \frac{20}{5} = \frac{4}{5} \]
Thus, the solution to the system of equations is: \[ (x, y) = \left(-\frac{4}{5}, \frac{4}{5}\right) \]
Make sure to graph the above points accurately to visualize the solution. If you have graphing software or tools, you can enter the equations and find the intersection point graphically as well.