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Chapter 3: Problem 54

Find the \(x\) - and \(y\) -intercepts. Then graph each equation. $$ 4 y=3 x $$

Short Answer

Expert verified
The x-intercept is (0, 0) and the y-intercept is (0, 0). To graph, plot (0, 0) and (4, 3), then draw a line through these points.

Step by step solution

01

- Find the x-intercept

To find the x-intercept, set y = 0 in the equation and solve for x. Given equation: 4y = 3x Substitute y = 0: 4(0) = 3x 0 = 3x Divide both sides by 3: x = 0 Thus, the x-intercept is (0, 0).
02

- Find the y-intercept

To find the y-intercept, set x = 0 in the equation and solve for y. Given equation: 4y = 3x Substitute x = 0: 4y = 3(0) 4y = 0 Divide both sides by 4: y = 0 Thus, the y-intercept is (0, 0).
03

- Graph the equation

To graph the equation, use the intercepts and an additional point. The intercepts are both (0, 0), so choose another value for x to find a corresponding y value.For example, let x = 4: 4y = 3(4) 4y = 12 Divide both sides by 4: y = 3 So, the point (4, 3) lies on the line. Plot the points (0, 0) and (4, 3) on the graph and draw a straight line through them.

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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 graph is where the line crosses the x-axis.
To find this point, we set y to 0 in the given equation because the y-coordinate of any point on the x-axis is always 0.
For the equation given, which is \(4y = 3x\), we substitute y = 0:

4(0) = 3x
0 = 3x
When solving for x, we divide both sides by 3:
x = 0

So, the x-intercept is (0, 0). This means the line touches the x-axis at the origin point.
y-intercept
The y-intercept is where the graph crosses the y-axis.
This means the x-coordinate is 0 at this point.
We find the y-intercept by setting x to 0 in the equation \(4y = 3x\):

4y = 3(0)
4y = 0
Solving for y, we divide both sides by 4:
y = 0

Thus, the y-intercept is also (0, 0). This shows that the line also crosses the y-axis at the origin.
graphing linear equations
Graphing linear equations involves plotting points and drawing a line through them.
Using the intercepts helps simplify this process.
To graph the equation \(4y = 3x\), we first plot the intercepts: (0,0) in this case.
Because both the x-intercept and y-intercept are at (0,0), we need another point to determine the direction of the line.
Let's choose x = 4:

4y = 3(4)
4y = 12
Dividing by 4:
y = 3

Now, we have another point (4, 3).
We plot these points on the graph and draw a line through them to represent the equation \(4y = 3x\).
This straight line shows all possible x-y pairs that satisfy the equation.
solving equations
Solving equations means finding the values of variables that make the equation true.
In the context of intercepts:
  • For the x-intercept, solve for x when y = 0.
  • For the y-intercept, solve for y when x = 0.
Let's break this down:
For \(4y = 3x\), we solve for x-intercept by setting y = 0 and isolating x:

4(0) = 3x
0 = 3x
x = 0

For the y-intercept, we set x = 0 and solve for y:

4y = 3(0)
4y = 0
y = 0

Both steps involve basic algebra: substituting zero for one variable and solving for the other.
This algebraic manipulation is a fundamental skill in determining intercepts and other solutions for equations.

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