To solve the system of equations by graphing, we'll first rewrite each equation in slope-intercept form, $y = mx + b$, and then plot the lines on a graph.

Step 1: Rewrite the Equations in Slope-Intercept Form

1. Equation 1: $x - y = 3$

   Rearrange to solve for $y$: $y = x - 3$

2. Equation 2: $7x - y = -3$

   Rearrange to solve for $y$: $y = 7x + 3$

Step 2: Plot the Lines on a Graph

• For the first equation $y = x - 3$:

  • The slope $m = 1$ and the y-intercept $b = -3$.
  • Start at $(0, -3)$ and use the slope to find another point: move up 1 unit and right 1 unit to reach $(1, -2)$.

• For the second equation $y = 7x + 3$:

  • The slope $m = 7$ and the y-intercept $b = 3$.
  • Start at $(0, 3)$ and use the slope to find another point: move up 7 units and right 1 unit to reach $(1, 10)$.

Step 3: Find the Intersection

The solution to the system of equations is the point where the two lines intersect on the graph. Let's calculate it directly for accuracy:

Solution

To find the intersection point:

y & = x - 3 \quad \text{(Equation 1)} \\ y & = 7x + 3 \quad \text{(Equation 2)} \end{aligned}$$ Set the two equations equal to each other: $$x - 3 = 7x + 3$$ Solve for $$x$$: $$x - 7x = 3 + 3 \implies -6x = 6 \implies x = -1$$ Now, substitute $$x = -1$$ into $$y = x - 3$$: $$y = -1 - 3 = -4$$ So, the lines intersect at $$(-1, -4)$$. **Graphically, the solution is the point $$(-1, -4)$$.** Would you like more details, or do you have any questions? Here are some related questions: 1. What are the slopes of both lines, and how do they compare? 2. How would you graph an equation if it were in a different form, such as standard form? 3. Can you solve this system using substitution or elimination instead? 4. What does it mean if the lines were parallel on the graph? 5. How can you check the solution algebraically after graphing? **Tip:** When graphing, always ensure you accurately plot points and check the scales on your axes to avoid errors.