To solve this system of equations by graphing, we need to graph each equation and identify the point where the two lines intersect. This intersection point represents the solution to the system of equations.

Step 1: Rewrite each equation in slope-intercept form ( $y = mx + b$ ).

1. Equation 1: $4x + 3y = 2$

   • Isolate $y$: $3y = -4x + 2$ $y = -\frac{4}{3}x + \frac{2}{3}$

   So, the first line has a slope of $-\frac{4}{3}$ and a y-intercept of $\frac{2}{3}$.

2. Equation 2: $3x - 2y = 10$

   • Isolate $y$: $-2y = -3x + 10$ $y = \frac{3}{2}x - 5$

   This gives us a slope of $\frac{3}{2}$ and a y-intercept of $-5$ for the second line.

Step 2: Plot each line

1. For the first equation $y = -\frac{4}{3}x + \frac{2}{3}$:

   • Start at $(0, \frac{2}{3})$ for the y-intercept.
   • Use the slope $-\frac{4}{3}$, which means go down 4 units and right 3 units to plot another point.

2. For the second equation $y = \frac{3}{2}x - 5$:

   • Start at $(0, -5)$ for the y-intercept.
   • Use the slope $\frac{3}{2}$, which means go up 3 units and right 2 units to plot another point.

Step 3: Find the Intersection Point

By graphing these lines, we see they intersect at the point $(4, -2)$.

Solution

The solution to the system is: $(x, y) = (4, -2)$

Let me know if you’d like a visual graph of the lines or if you have more questions!

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Relative Questions

1. How do you verify the solution by substituting it back into the original equations?
2. Can this system be solved using another method, like substitution or elimination?
3. What does it mean if two lines are parallel in terms of their slopes and solutions?
4. How can you determine the type of solution (one solution, no solution, or infinitely many) by looking at the slopes?
5. What role does the y-intercept play in graphing these lines?

Tip

When graphing, always start from the y-intercept, then use the slope to find additional points for accuracy.