Graph x + 3y = -9 and 5x + 3y = 3 with points to graph and y intersection

To graph the system of equations:

1. \( x + 3y = -9 \)
2. \( 5x + 3y = 3 \)

### Step 1: Solve for \( y \) in both equations.
Let's isolate \( y \) in both equations to find points and the y-intercept for graphing.

**Equation 1:**
\[ x + 3y = -9 \]
Solve for \( y \):
\[
3y = -x - 9
\]
\[
y = -\frac{1}{3}x - 3
\]
So the slope is \( -\frac{1}{3} \) and the y-intercept is \( -3 \) (i.e., point \( (0, -3) \)).

**Equation 2:**
\[ 5x + 3y = 3 \]
Solve for \( y \):
\[
3y = -5x + 3
\]
\[
y = -\frac{5}{3}x + 1
\]
So the slope is \( -\frac{5}{3} \) and the y-intercept is \( 1 \) (i.e., point \( (0, 1) \)).

### Step 2: Find points to graph.

We can use a few values of \( x \) to find corresponding points on each line.

**For \( y = -\frac{1}{3}x - 3 \):**
- If \( x = 0 \), \( y = -3 \). So, point \( (0, -3) \).
- If \( x = 3 \), \( y = -4 \). So, point \( (3, -4) \).
- If \( x = -3 \), \( y = -2 \). So, point \( (-3, -2) \).

**For \( y = -\frac{5}{3}x + 1 \):**
- If \( x = 0 \), \( y = 1 \). So, point \( (0, 1) \).
- If \( x = 3 \), \( y = -4 \). So, point \( (3, -4) \).
- If \( x = -3 \), \( y = 6 \). So, point \( (-3, 6) \).

### Step 3: Plot points and draw the lines.

- For the first line \( y = -\frac{1}{3}x - 3 \), plot the points \( (0, -3), (3, -4), (-3, -2) \).
- For the second line \( y = -\frac{5}{3}x + 1 \), plot the points \( (0, 1), (3, -4), (-3, 6) \).

### Step 4: Intersection point.

Both lines pass through \( (3, -4) \), so this is the solution to the system of equations.

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### 5 Related Questions:
1. How do we find the solution to a system of linear equations using substitution or elimination?
2. What is the significance of the slope in a linear equation?
3. How can we determine if two lines are parallel, perpendicular, or intersecting?
4. What does the y-intercept represent in a real-world context?
5. How can you interpret the solution of a system of equations geometrically?

### Tip:
Remember that when two lines intersect, their point of intersection is the solution to the system of equations.

Learn how to graph the system of linear equations x + 3y = -9 and 5x + 3y = 3 by finding the slope, y-intercept, and points for each equation. Understand how to identify the intersection point and solve the system. Suitable for students in Grades 9-10.

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