Question

(a) find the intercepts of the graph of each equation and (b) graph the equation. $$ x-\frac{2}{3} y=4 $$

Short answer

(4,0) and (0,-6) are intercepts. Graph the points and connect them.

Step-by-step solution

Find the x-intercept

To find the x-intercept, set y = 0 in the equation and solve for x. The equation is: \text{ } \[ x - \frac{2}{3} y = 4 \]. \text{ } Substitute y = 0: \text{ } \[ x - \frac{2}{3} (0) = 4 \]. \text{ } This simplifies to x = 4. Therefore, the x-intercept is (4, 0).

Find the y-intercept

To find the y-intercept, set x = 0 in the equation and solve for y. The equation is: \text{ } \[ x - \frac{2}{3} y = 4 \]. \text{ } Substitute x = 0: \text{ } \[ 0 - \frac{2}{3} y = 4 \]. \text{ } Simplify the equation: \[ - \frac{2}{3} y = 4 \]. Multiply both sides by -3/2 to solve for y: \[ y = -6 \]. Therefore, the y-intercept is (0, -6).

Create a table of values

Choose additional x-values and solve for y to get more points for graphing. For instance, let x = 2:\text{ } \[ 2 - \frac{2}{3} y = 4 \]. Subtract 2 from both sides: \text{ } \[ - \frac{2}{3} y = 2 \]. Multiply both sides by -3/2: \text{ } \[ y = -3 \]. So, (2, -3) is another point on the line.

Graph the equation

Plot the intercepts (4, 0) and (0, -6) on a coordinate plane. Also, plot the point (2, -3). Draw a straight line through these points to graph the equation.

Key concepts

x-intercept

The x-intercept of a line is the point where the line crosses the x-axis. To find this point, you set the value of y to zero in the equation of the line and solve for x. This works because any point on the x-axis has a y-coordinate of 0.

In our example, the equation is \[ x - \frac{2}{3} y = 4 \].

By setting y to 0, it simplifies to \[ x - \frac{2}{3} \cdot 0 = 4 \], which then becomes \[ x = 4 \].

Therefore, the x-intercept is at the point (4, 0).

y-intercept

The y-intercept is the point where the line crosses the y-axis. To find this point, set the value of x to zero and solve for y. This works because any point on the y-axis has an x-coordinate of 0.

In our equation \[ x - \frac{2}{3} y = 4 \], we set x to 0, giving us \[ 0 - \frac{2}{3} y = 4 \].

This simplifies to \[ - \frac{2}{3} y = 4 \].

To isolate y, multiply both sides by -3/2: \[ y = -6 \].

Therefore, the y-intercept is at (0, -6).

graphing lines

Graphing a linear equation involves plotting points and drawing a straight line through them.

Start by plotting the x-intercept (4, 0) and the y-intercept (0, -6).

It's also helpful to find additional points for accuracy. For example, if x = 2, then: \[ 2 - \frac{2}{3} y = 4 \]

Simplify to \[ -\frac{2}{3} y = 2 \] and solve for y by multiplying both sides by -3/2, which gives \[ y = -3 \].

This gives us the point (2, -3).

Plot all these points on a coordinate grid and connect them with a straight line to complete the graph.

solving equations

Solving linear equations is a key skill. The goal is to find the values of x and y that make the equation true.

Follow these steps:

• Isolate the variable: If solving for x, you want it alone on one side of the equation.


• Use inverse operations: Add, subtract, multiply, or divide both sides to isolate the variable.


• Check your solution: Substitute back into the original equation to ensure it works.

For example, to find the y-intercept in our equation \[ x - \frac{2}{3} y = 4 \], we set x to 0, giving \[ -\frac{2}{3} y = 4 \].

Solving for y involves multiplying both sides by -3/2, giving \[ y = -6 \].

Always verify by substituting back into the original equation!