Solve the system by graphing. 2x+3y=9; x-2y=8

To solve the system of equations by graphing, follow these steps:

### 1. Rewrite both equations in slope-intercept form (y = mx + b).

#### For the first equation:
\[
2x + 3y = 9
\]
Solve for \( y \):
\[
3y = -2x + 9
\]
\[
y = -\frac{2}{3}x + 3
\]

#### For the second equation:
\[
x - 2y = 8
\]
Solve for \( y \):
\[
-2y = -x + 8
\]
\[
y = \frac{1}{2}x - 4
\]

### 2. Graph both lines.

- For the first line \( y = -\frac{2}{3}x + 3 \):
  - **Slope**: \( -\frac{2}{3} \)
  - **y-intercept**: \( 3 \) (the point \( (0, 3) \))

- For the second line \( y = \frac{1}{2}x - 4 \):
  - **Slope**: \( \frac{1}{2} \)
  - **y-intercept**: \( -4 \) (the point \( (0, -4) \))

### 3. Find the intersection point by graphing both lines and checking where they cross.

Plot both lines using their slopes and y-intercepts, and the point where the two lines intersect is the solution to the system.

### 4. Algebraic Solution for Intersection Point

Instead of graphing, you can solve algebraically:
1. Set the two equations for \( y \) equal to each other:
   \[
   -\frac{2}{3}x + 3 = \frac{1}{2}x - 4
   \]
2. Solve for \( x \):
   \[
   -\frac{2}{3}x - \frac{1}{2}x = -4 - 3
   \]
   \[
   -\frac{4}{6}x - \frac{3}{6}x = -7
   \]
   \[
   -\frac{7}{6}x = -7
   \]
   \[
   x = 6
   \]

3. Substitute \( x = 6 \) into one of the original equations to find \( y \):
   Using \( y = \frac{1}{2}x - 4 \):
   \[
   y = \frac{1}{2}(6) - 4 = 3 - 4 = -1
   \]

### Solution:
The lines intersect at \( (6, -1) \), so the solution to the system is:
\[
x = 6, \quad y = -1
\]

Do you have any questions, or would you like further details?

#### 5 Related Questions:
1. How do you determine the slope of a line from its equation?
2. What is the significance of the y-intercept in graphing a line?
3. How can substitution or elimination methods be used instead of graphing to solve a system of equations?
4. How do you convert between different forms of linear equations?
5. What does it mean when two lines in a system are parallel?

#### Tip:
When graphing, always label the points where the lines cross the axes to ensure accuracy in locating the intersection.

Learn how to solve the system of equations 2x + 3y = 9 and x - 2y = 8 by graphing. This problem explains key concepts like slope-intercept form, graphing lines, and finding their intersection point. Suitable for students in Grades 8-10.

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