Problem 15 - Understanding Elementary Algebra with Geometry

Question

Find the \(x\) - and \(y\) -intercepts of the equation. $$y=4 x$$

Step-by-Step Solution

Answer

Both the x-intercept and y-intercept are (0, 0).

Step 1: Find the y-intercept

To find the y-intercept, set (x = 0) in the equation y = 4x. Substituting x = 0 into y = 4x, we get y = 4(0) = 0. Therefore, the y-intercept is (0, 0).

Step 2: Find the x-intercept

To find the x-intercept, set (y = 0) in the equation y = 4x. Substituting y = 0 into y = 4x, we get 0 = 4x. Now, solve for x by dividing both sides by 4: x = 0. Therefore, the x-intercept is (0, 0).

Step 3: Summary of intercepts

Both the x-intercept and y-intercept are located at (0, 0). This means the line intersects both the x-axis and y-axis at the origin.

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, set the y-value to 0 and solve for x.
As seen in the original exercise, start by taking the equation: \[ y = 4x \]. Set y to 0 and solve for x:
\[ 0 = 4x \]
Divide both sides by 4:
\[ x = 0 \]
So, the x-intercept of the equation is the point \((0, 0)\).
Remember, the importance of the x-intercept is to determine where the line crosses the x-axis, which can be particularly useful in graphing the equation or understanding its behavior.

y-intercept

The y-intercept is the point where the graph of a linear equation crosses the y-axis. To find the y-intercept, set the x-value to 0 and solve for y.
Using the same equation as before: \[ y = 4x \]. Set x to 0 and solve for y:
\[ y = 4(0) = 0 \]
So, the y-intercept of the equation is the point \((0, 0)\).
This means that when x is 0, y is also 0. The y-intercept is an essential concept because it shows where the line crosses the y-axis, helping us to visualize the line's position relative to the origin.

linear equations

Linear equations are algebraic expressions that represent straight lines when graphed on a coordinate plane. These equations typically take the form \( y = mx + b \), where m is the slope and b is the y-intercept.
For the equation \( y = 4x \), it can be rewritten in this form as \( y = 4x + 0 \), where:

• m (slope) = 4: This indicates the line rises 4 units for every 1 unit it moves horizontally.
• b (y-intercept) = 0: This tells us the line crosses the y-axis at 0.

Understanding linear equations allows us to predict and calculate the y-value for any given x-value, which is fundamental in algebra and many real-world applications. Whether you're graphing the line or using it in a word problem, knowing how to work with linear equations is crucial.