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graphing linear equations in Two Variables using x – and y – intercepts

To solve the system of linear equations by graphing, we'll follow these steps:

### Given Equations:
1. \( 4x + 3y = 12 \)
2. \( y = 2x + 1 \)

### Step 1: Find the x- and y-intercepts of the first equation \( 4x + 3y = 12 \).

- **For the x-intercept**: Set \( y = 0 \) and solve for \( x \).
  \[
  4x + 3(0) = 12 \implies 4x = 12 \implies x = 3
  \]
  So, the x-intercept is \( (3, 0) \).

- **For the y-intercept**: Set \( x = 0 \) and solve for \( y \).
  \[
  4(0) + 3y = 12 \implies 3y = 12 \implies y = 4
  \]
  So, the y-intercept is \( (0, 4) \).

### Step 2: Plot the second equation \( y = 2x + 1 \).

This is already in slope-intercept form \( y = mx + b \), where:
- \( m = 2 \) (slope), and
- \( b = 1 \) (y-intercept).

So, the y-intercept is \( (0, 1) \), and the slope indicates that for every increase of 1 in \( x \), \( y \) increases by 2.

### Step 3: Graphing both equations

1. For the first equation \( 4x + 3y = 12 \), plot the points \( (3, 0) \) and \( (0, 4) \), then draw a line through these points.
2. For the second equation \( y = 2x + 1 \), plot the point \( (0, 1) \) and use the slope \( 2 \) to find another point, e.g., \( (1, 3) \) (since the slope is 2, when \( x \) increases by 1, \( y \) increases by 2). Draw a line through these points.

### Step 4: Intersection
The point where the two lines intersect is the solution to the system of equations. This point satisfies both equations.

Do you want details on how to graph the system or any other specific part explained? Here are some related questions:

1. How can I calculate the slope of a line from two points?
2. Can you explain how to graph lines from slope-intercept form?
3. How do you check if two lines are parallel or perpendicular?
4. What does it mean when two lines have no intersection?
5. How can I solve this system algebraically instead of graphing?

**Tip:** To accurately graph, you can use graph paper or an online graphing tool for better precision in finding the intersection point.

Graphing linear equations in two variables using x- and y-intercepts: Solve the system of equations 4x + 3y = 12 and y = 2x + 1.

Learn how to solve the system of linear equations 4x + 3y = 12 and y = 2x + 1 by graphing. This step-by-step guide includes finding intercepts and plotting the equations for Grades 8-10.

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