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.