Problem 14 - Understanding Elementary Algebra with Geometry

Question

Find the \(x\) - and \(y\) -intercepts of the equation. $$3 x+2 y=12$$

Step-by-Step Solution

Answer

The x-intercept is (4, 0) and the y-intercept is (0, 6).

Step 1: Identify the x-intercept

To find the x-intercept of the equation, set y to 0 and solve for x. The x-intercept is where the line crosses the x-axis.

Step 2: Substitute y = 0

Substitute 0 for y in the equation:\[3x + 2(0) = 12\]Simplifies to:\[3x = 12\]

Step 3: Solve for x

Solve the equation for x by dividing both sides by 3:\[x = \frac{12}{3}\]\[x = 4\]Therefore, the x-intercept is (4, 0).

Step 4: Identify the y-intercept

To find the y-intercept of the equation, set x to 0 and solve for y. The y-intercept is where the line crosses the y-axis.

Step 5: Substitute x = 0

Substitute 0 for x in the equation:\[3(0) + 2y = 12\]Simplifies to:\[2y = 12\]

Step 6: Solve for y

Solve the equation for y by dividing both sides by 2:\[y = \frac{12}{2}\]\[y = 6\]Therefore, the y-intercept is (0, 6).

Key Concepts

x-intercept

The x-intercept of a linear equation is the point where the graph of the equation crosses the x-axis. To find the x-intercept, we set the value of y to 0 in the equation because at the x-intercept, y is always 0.

Follow these steps to find the x-intercept:

• Identify the equation of the line. In our example, it is: \(3x + 2y = 12\).
• Set y to 0 because we are finding the point where the line crosses the x-axis.
• Substitute y = 0 into the equation.
• Solve the resulting equation for x.

For the equation \(3x + 2y = 12\), we substitute y = 0 to get: \(3x + 2(0) = 12\), which simplifies to \(3x = 12\).

Now, solve for x by dividing both sides of the equation by 3:
\[x = \frac{12}{3} \]
\[x = 4 \]
Thus, the x-intercept is \((4, 0)\). This means the line crosses the x-axis at the point (4, 0).

y-intercept

The y-intercept of a linear equation is the point where the graph crosses the y-axis. To find the y-intercept, we set the value of x to 0 because at the y-intercept, x is always 0.

Here are the steps to determine the y-intercept:

• Start with the equation of the line, which in our example is \(3x + 2y = 12\).
• Set x to 0 because we are finding the point where the line crosses the y-axis.
• Substitute x = 0 into the equation.
• Solve the resulting equation for y.

For the equation \(3x + 2y = 12\), we set x to 0 to get: \(3(0) + 2y = 12\), which simplifies to \(2y = 12\).

Next, solve for y by dividing both sides of the equation by 2:
\[y = \frac{12}{2} \]
\[y = 6 \]
Therefore, the y-intercept is \((0, 6)\). This means the line crosses the y-axis at the point (0, 6).

linear equations

A linear equation is an equation that forms a line when it is graphed on a coordinate plane. These equations can be written in various forms, with one of the most common being the slope-intercept form: \(y = mx + b\), where m represents the slope and b represents the y-intercept.

For our example, the linear equation is given in standard form: \(3x + 2y = 12\).

• Linear equations create straight lines on the graph.
• Each linear equation has an x-intercept and a y-intercept, where the line crosses the x-axis and y-axis respectively.
• Intercepts help us understand where the line is positioned relative to the axes.

To understand linear equations better, remember:

• The x-intercept is found by setting y = 0.
• The y-intercept is found by setting x = 0.
• Both intercepts provide key points on the line, making it easier to plot the graph.

Understanding and finding intercepts is fundamental in mastering linear equations and graphing lines accurately.