Chapter 2: Problem 106
(a) find the intercepts of the graph of each equation and (b) graph the equation. $$ x-\frac{2}{3} y=4 $$
Short Answer
Expert verified
(4,0) and (0,-6) are intercepts. Graph the points and connect them.
Step by step solution
01
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).
02
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).
03
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.
04
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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).
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).
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.
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:
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!
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!