To solve the given system of equations using a graphical method, we need to graph each equation and identify the point where the two lines intersect. This point of intersection will give us the solution to the system of equations. Let's follow these steps in detail:
1. Rewrite the Equations in Slope-Intercept Form:
- The slope-intercept form of a line is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
For the first equation:
[tex]\[
4x + 9y = 12
\][/tex]
Solve for [tex]\( y \)[/tex]:
[tex]\[
9y = -4x + 12
\][/tex]
[tex]\[
y = -\frac{4}{9}x + \frac{12}{9}
\][/tex]
Simplify the constant term:
[tex]\[
y = -\frac{4}{9}x + \frac{4}{3}
\][/tex]
For the second equation:
[tex]\[
-2x + y = -6
\][/tex]
Solve for [tex]\( y \)[/tex]:
[tex]\[
y = 2x - 6
\][/tex]
2. Graph the First Equation [tex]\( y = -\frac{4}{9}x + \frac{4}{3} \)[/tex]:
- The y-intercept ([tex]\( b \)[/tex]) is [tex]\( \frac{4}{3} \approx 1.33 \)[/tex], so plot the point (0, [tex]\( \frac{4}{3} \)[/tex]).
- The slope ([tex]\( m \)[/tex]) is [tex]\( -\frac{4}{9} \)[/tex], which means for every 9 units you move right on the x-axis, you move 4 units down on the y-axis.
- Starting from the y-intercept, move right by 9 units and down by 4 units to plot another point.
- Draw the line through these points.
3. Graph the Second Equation [tex]\( y = 2x - 6 \)[/tex]:
- The y-intercept ([tex]\( b \)[/tex]) is [tex]\(-6\)[/tex], so plot the point (0, -6).
- The slope ([tex]\( m \)[/tex]) is [tex]\( 2 \)[/tex], which means for every 1 unit you move right on the x-axis, you move 2 units up on the y-axis.
- Starting from the y-intercept, move right by 1 unit and up by 2 units to plot another point.
- Draw the line through these points.
4. Find the Intersection of the Two Lines:
- Plot both lines on the same graph.
- Observe the point where the two lines intersect. This intersection point is the solution to the system of equations.
After graphing these equations, we can see that the intersection point is at [tex]\( (3, 0) \)[/tex].
So, the solution to the system of equations is:
[tex]\[
x = 3, \quad y = 0
\][/tex]
which means the solution is the point [tex]\( (3, 0) \)[/tex].
