Solve the following system of linear equation graphically 2x + 1 = y , x + 3y = 15

To solve the system of linear equations graphically, we need to:

1. Express each equation in the slope-intercept form (\(y = mx + b\)).
2. Plot both equations on the same coordinate plane.
3. Find the point where the two lines intersect, which will be the solution to the system of equations.

### Step 1: Rearranging the equations

#### First equation: \( 2x + 1 = y \)
- This is already in slope-intercept form: 
  \[
  y = 2x + 1
  \]

#### Second equation: \( x + 3y = 15 \)
- Solve for \(y\):
  \[
  3y = 15 - x
  \]
  \[
  y = \frac{15 - x}{3} = -\frac{1}{3}x + 5
  \]

So the two equations to graph are:
1. \( y = 2x + 1 \)
2. \( y = -\frac{1}{3}x + 5 \)

### Step 2: Plotting the equations

For the first equation \( y = 2x + 1 \):
- The y-intercept is \(1\) (the point \( (0, 1) \)).
- The slope is \(2\), meaning for every 1 unit increase in \(x\), \(y\) increases by 2.

For the second equation \( y = -\frac{1}{3}x + 5 \):
- The y-intercept is \(5\) (the point \( (0, 5) \)).
- The slope is \( -\frac{1}{3} \), meaning for every 3 units increase in \(x\), \(y\) decreases by 1.

### Step 3: Finding the intersection

The intersection point is the solution to the system of equations. Solving algebraically to verify:

Set the equations equal to each other:
\[
2x + 1 = -\frac{1}{3}x + 5
\]
Multiply both sides by 3 to eliminate the fraction:
\[
6x + 3 = -x + 15
\]
Add \(x\) to both sides:
\[
7x + 3 = 15
\]
Subtract 3 from both sides:
\[
7x = 12
\]
Divide by 7:
\[
x = \frac{12}{7}
\]

Substitute \(x = \frac{12}{7}\) into \(y = 2x + 1\):
\[
y = 2\left(\frac{12}{7}\right) + 1 = \frac{24}{7} + 1 = \frac{24}{7} + \frac{7}{7} = \frac{31}{7}
\]

Thus, the solution to the system is:
\[
\left( \frac{12}{7}, \frac{31}{7} \right)
\]

Would you like me to provide a graph of the solution?

### Relative Questions:
1. How do you graph linear equations with fractional slopes?
2. What does it mean if two lines are parallel in a system of equations?
3. Can you solve a system of equations using substitution or elimination methods?
4. What is the significance of the point of intersection in a system of equations?
5. How would you solve a system of nonlinear equations graphically?

**Tip:** In systems of equations, the point of intersection represents the solution where both equations hold true simultaneously.

Solve the following system of linear equations graphically: 2x + 1 = y , x + 3y = 15

Learn how to solve the system of linear equations 2x + 1 = y and x + 3y = 15 graphically. This detailed guide walks you through rearranging equations into slope-intercept form, plotting them on a coordinate plane, and finding the intersection point. Suitable for Grades 9-11 students.

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