Question

(a) find the intercepts of the graph of each equation and (b) graph the equation. $$ \frac{1}{2} x+\frac{1}{3} y=1 $$

Short answer

Intercepts: (2, 0) and (0, 3). Graph: Draw a line through these points.

Step-by-step solution

- Find the x-intercept

To find the x-intercept, set y to 0 in the equation and solve for x. The equation becomes: \[ \frac{1}{2} x + \frac{1}{3} \times 0 = 1 \] This simplifies to: \[ \frac{1}{2} x = 1 \] Multiply both sides by 2 to solve for x: \[ x = 2 \] So, the x-intercept is at (2, 0).

- Find the y-intercept

To find the y-intercept, set x to 0 in the equation and solve for y. The equation becomes: \[ \frac{1}{2} \times 0 + \frac{1}{3} y = 1 \] This simplifies to: \[ \frac{1}{3} y = 1 \] Multiply both sides by 3 to solve for y: \[ y = 3 \] So, the y-intercept is at (0, 3).

- Graph the equation

With the intercepts from the previous steps, plot the points (2, 0) and (0, 3) on a coordinate plane. Draw a straight line through these points, and extend it in both directions. This is the graph of the equation \( \frac{1}{2} x + \frac{1}{3} y = 1 \).

Key concepts

Intercepts

Intercepts are where the graph of an equation crosses the axes on a coordinate plane. There are two types of intercepts:

• x-intercept: This is the point where the graph crosses the x-axis. To find it, set y to 0 in the equation and solve for x.
• y-intercept: This is the point where the graph crosses the y-axis. To find it, set x to 0 in the equation and solve for y.

For example, in the equation \[ \frac{1}{2} x + \frac{1}{3} y = 1 \], you can find the intercepts like this:

1. **Find the x-intercept**: Set y = 0 and solve for x. The equation simplifies to \[ \frac{1}{2} x = 1 \], giving the x-intercept (2, 0).
2. **Find the y-intercept**: Set x = 0 and solve for y. The equation simplifies to \[ \frac{1}{3} y = 1 \], giving the y-intercept (0, 3).

Graphing

Graphing a linear equation involves plotting points on a coordinate plane and then drawing a line through them. Here's a simple way to graph the equation \[ \frac{1}{2} x + \frac{1}{3} y = 1 \]:

• Step 1: Find the intercepts (as shown in the previous section). The intercepts are (2, 0) and (0, 3).
• Step 2: Plot the intercepts on the coordinate plane. Mark the points (2, 0) on the x-axis and (0, 3) on the y-axis.
• Step 3: Draw a straight line connecting these two points. Extend the line in both directions.

This line represents all the solutions of the given linear equation. Any point on this line is a solution to the equation \[ \frac{1}{2} x + \frac{1}{3} y = 1 \].

Solving Equations

To solve a linear equation means to find the values of x and y that make the equation true. There are a few methods, but let's focus on solving for intercepts as shown in our example:

• Step 1: Solve for x-intercept. Set y = 0 and solve for x. For \[ \frac{1}{2} x + \frac{1}{3} y = 1 \], setting y to 0 gives \[ \frac{1}{2} x = 1 \]. Multiply by 2 to get x = 2.
• Step 2: Solve for y-intercept. Set x = 0 and solve for y. For \[ \frac{1}{2} x + \frac{1}{3} y = 1 \], setting x to 0 gives \[ \frac{1}{3} y = 1 \]. Multiply by 3 to get y = 3.
• Step 3: Check your solutions. Substitute back into the original equation to ensure accuracy. For x=2, y=0: \[ \frac{1}{2} \times 2 + \frac{1}{3} \times 0 = 1 \] checks out. For x=0, y=3: \[ \frac{1}{2} \times 0 + \frac{1}{3} \times 3 = 1 \] also checks out.

Understanding these steps ensures you can always find the intercepts and graph any linear equation correctly.