Let's solve this system of linear equations and match it with the graph provided.

The equations are:

1. $y = \frac{2}{3}x - 3$
2. $y = -2x + 5$

Step 1: Graphing the first equation

The equation $y = \frac{2}{3}x - 3$ is in slope-intercept form, $y = mx + b$, where:

• $m = \frac{2}{3}$ (slope)
• $b = -3$ (y-intercept)

To graph this:

• The line passes through $(0, -3)$ on the y-axis.
• The slope $\frac{2}{3}$ means for every 3 units moved to the right, the line moves up by 2 units.

Step 2: Graphing the second equation

The second equation $y = -2x + 5$ is also in slope-intercept form, where:

• $m = -2$ (slope)
• $b = 5$ (y-intercept)

To graph this:

• The line passes through $(0, 5)$ on the y-axis.
• The slope $-2$ means for every 1 unit moved to the right, the line moves down by 2 units.

Step 3: Finding the intersection point

Now, solving these equations algebraically to find the intersection (solution):

Set the two equations equal to each other: $\frac{2}{3}x - 3 = -2x + 5$ To eliminate the fraction, multiply the entire equation by 3: $2x - 9 = -6x + 15$ Now, combine like terms: $2x + 6x = 15 + 9$ $8x = 24$ Solve for $x$: $x = 3$

Substitute $x = 3$ back into one of the original equations (I'll use $y = -2x + 5$): $y = -2(3) + 5 = -6 + 5 = -1$

Thus, the solution is $(3, -1)$, meaning the two lines intersect at the point $(3, -1)$.

Matching with the graph:

From the graph, we observe that the lines intersect at approximately $(3, -1)$, confirming that the graph corresponds to this system of linear equations.

Would you like more details or have any questions?

Here are 5 questions related to this:

1. How do you find the slope of a line from an equation?
2. What is the graphical meaning of the slope in a linear equation?
3. Can two lines have the same slope but different y-intercepts? What happens in this case?
4. How do you solve a system of equations by substitution or elimination?
5. What are the conditions for a system of linear equations to have no solution?

Tip: When graphing a linear equation, always start by plotting the y-intercept and then use the slope to plot additional points.