To solve the system of equations by graphing, we'll first create a T-chart for each equation to find two pairs of coordinates. Then, we'll use these coordinates to graph the lines and find their intersection, which represents the solution to the system.

Equation 1: $x + 2y = 6$

Step 1: Solve for $y$

To make it easier to fill in the T-chart, solve the equation for $y$: $2y = 6 - x$ $y = \frac{6 - x}{2}$

Step 2: Create the T-Chart

Let's choose two values for $x$ and find the corresponding $y$ values.

$x$	$y$
0	$y = \frac{6 - 0}{2} = 3$
2	$y = \frac{6 - 2}{2} = 2$

Coordinates:

• (0, 3)
• (2, 2)

Equation 2: $2x - 3y = 12$

Step 1: Solve for $y$

Similarly, solve the second equation for $y$: $-3y = 12 - 2x$ $y = \frac{2x - 12}{3}$

Step 2: Create the T-Chart

Choose two values for $x$ and find the corresponding $y$ values.

$x$	$y$
0	$y = \frac{2(0) - 12}{3} = -4$
3	$y = \frac{2(3) - 12}{3} = -2$

Coordinates:

• (0, -4)
• (3, -2)

Step 3: Graph the Equations

1. For $x + 2y = 6$:

   • Plot the points (0, 3) and (2, 2).
   • Draw a straight line through these points.

2. For $2x - 3y = 12$:

   • Plot the points (0, -4) and (3, -2).
   • Draw a straight line through these points.

Step 4: Find the Intersection

• The intersection point of the two lines is the solution to the system of equations.

Solution: The lines intersect at the point $(6, 0)$.

Would you like to see the graph or need further details?

Related Questions:

1. How do you convert equations to slope-intercept form?
2. What are other methods to solve a system of equations besides graphing?
3. How do you verify if the intersection point is a solution to both equations?
4. What happens if you get a different pair of points to plot the same line?
5. How does changing the coefficients affect the graph of the equation?

Tip:

Always check the intersection point by substituting it back into both original equations to confirm it's a valid solution.