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Chapter 3: Problem 44

In the following exercises, find the intercepts for each equation.. $$ 3 x-2 y=12 $$

Short Answer

Expert verified
x-intercept: (4, 0), y-intercept: (0, -6)

Step by step solution

01

Identify the x-intercept

To find the x-intercept, set y to 0 in the equation and solve for x. Start with the equation: 3x - 2(0) = 12 3x = 12 Divide both sides by 3: x = 4 x-intercept is (4, 0)
02

Identify the y-intercept

To find the y-intercept, set x to 0 in the equation and solve for y. Start with the equation: 3(0) - 2y = 12 -2y = 12 Divide both sides by -2: y = -6 y-intercept is (0, -6)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

x-intercept
An x-intercept is where a graph crosses the x-axis. This point has a y-coordinate of zero because it lies on the x-axis. To find it, we set y to zero in the equation and solve for x.
For the equation 3x - 2y = 12, set y to 0:
3x - 2(0) = 12
3x = 12
Now, simply divide both sides by 3:
x = 4
So, the x-intercept is at (4, 0). This means the line crosses the x-axis at the point (4, 0).
Understanding x-intercepts helps us know where our graph touches the x-axis. It's crucial for sketching graphs and understanding linear equations.
y-intercept
A y-intercept is where a graph crosses the y-axis. This point has an x-coordinate of zero because it lies on the y-axis. To find it, set x to zero in the equation and solve for y.
For the equation 3x - 2y = 12, set x to 0:
3(0) - 2y = 12
-2y = 12
Now, divide both sides by -2:
y = -6
So, the y-intercept is at (0, -6). This means the line crosses the y-axis at the point (0, -6).
Knowing y-intercepts helps us determine where our graph touches the y-axis, which is essential for plotting accurate graphs and solving linear equations efficiently.
solving equations
Solving equations means finding the values of variables that make the equation true. There are various methods to solve equations, such as:
  • Isolating the variable
  • Using algebraic manipulations
  • Graphing
To isolate the variable, we perform operations that will get the variable by itself on one side of the equation.
For example, in the equation 3x - 2y = 12, to find the x-intercept, we isolate x:
Set y to 0: 3x - 2(0) = 12
We then solve for x by dividing both sides by 3: x = 4
The same approach is applied for finding the y-intercept:
Set x to 0: 3(0) - 2y = 12
Solve for y by dividing both sides by -2: y = -6
Solving equations is a key skill in algebra and helps us understand the relationships between variables.
linear equations
Linear equations are equations of the first degree. This means they form a straight line when graphed. Their general form is:
ax + by = c
where a, b, and c are constants.
Let's look at the linear equation 3x - 2y = 12:
This equation can describe many real-world situations, such as budgeting, physics, or engineering problems.
To understand it better, we can find both the x- and y-intercepts.
Set y = 0 to find x:
3x = 12, so x = 4
Set x = 0 to find y:
-2y = 12, so y = -6
These two intercepts help us plot the straight line on a graph.
Linear equations are fundamental in algebra and serve as a building block for more advanced topics in mathematics.

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Most popular questions from this chapter

In the following exercises, (a) graph each function (b) state its domain and range. Write the domain and range in interval notation. $$ f(x)=\frac{1}{2} x^{2} $$

In the following exercises, (a) graph each function (b) state its domain and range. Write the domain and range in interval notation. $$ f(x)=-x-2 $$

Graph the linear inequality: \(4 x-3 y>12\)

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