Problem 13 - Understanding Elementary Algebra with Geometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the equation

The given equation is \(2x + 3y = 12\). This is a linear equation in two variables.

Step 2: Find the x-intercept

To find the \(x\)-intercept, set \(y = 0\) and solve for \(x\).\[2x + 3(0) = 12\]\[2x = 12\]\[x = 6\]So, the \(x\)-intercept is \( (6, 0) \).

Step 3: Find the y-intercept

To find the \(y\)-intercept, set \(x = 0\) and solve for \(y\).\[2(0) + 3y = 12\]\[3y = 12\]\[y = 4\]So, the \(y\)-intercept is \( (0, 4) \).

Key Concepts

x-intercept

Understanding the x-intercept of a linear equation is easy and important. You are essentially finding the point where the line crosses the x-axis. At the x-intercept, the value of y is always 0.
To find the x-intercept for the given equation, follow these steps:

• Set y to 0 in the equation.
• Solve the resulting equation for x.

For example, in the equation: \(2x + 3y = 12\), set y to 0:
\[2x + 3(0) = 12\]

Solving this, we get:
\[2x = 12\]
\[x = 6\]
Therefore, the x-intercept is (6, 0). This tells you that at the point where the line crosses the x-axis, the x-coordinate is 6.

y-intercept

The y-intercept is where the line crosses the y-axis. Here, the value of x is always 0. Finding the y-intercept involves a similar process to finding the x-intercept.
Follow these steps to find the y-intercept for any linear equation:

• Set x to 0 in the equation.
• Solve the resulting equation for y.

Using the given equation: \(2x + 3y = 12\), set x to 0:
\[2(0) + 3y = 12\]

Simplify and solve for y:
\[3y = 12\]
\[y = 4\]
Therefore, the y-intercept is (0, 4). This means that the point where the line crosses the y-axis has a y-coordinate of 4.

solving linear equations

Solving linear equations is a fundamental skill in algebra. A linear equation is any equation that can be written in the form \(Ax + By = C\), where A, B, and C are constants.
To solve a linear equation, you need to isolate the variable of interest. This often involves multiple steps:

• Combine like terms.
• Use addition or subtraction to move terms from one side of the equation to the other.
• Use multiplication or division to solve for the variable.

Let's break down the process using the given equation \(2x + 3y = 12\):

To find the x-intercept, set y to 0.
This gives \[2x = 12\] then solving for x, \[x = 6\]

To find the y-intercept, set x to 0.
This gives \[3y = 12\] then solving for y, \[y = 4\]

Remember, the key steps include:

• Substitute known values.
• Simplify to isolate the variable.
• Check your work by substituting the solution back into the original equation.

With practice, solving linear equations becomes second nature!