Problem 59 - Algebra 1

Question

Find the x-intercept and the y-intercept of the graph of the equation. $$x-1.1 y=10$$

Step-by-Step Solution

Answer

After solving, the x-intercept is \(x = 10\) and the y-intercept is \(y = - \frac{10}{1.1}\).

Step 1: Calculate the x-intercept

To find the x-intercept, set \(y = 0\) in the given equation and solve for \(x\). The equation is \(x - 1.1*0 = 10\). Solving this will give us the x-intercept.

Step 2: Calculate the y-intercept

To find the y-intercept, make \(x = 0\) in the given equation and solve for \(y\). So the equation becomes \(0 - 1.1*y = 10\). Finding the solution for \(y\) will give the y-intercept.

Key Concepts

Understanding the X-Intercept

The x-intercept of a linear equation is an essential concept. It is the point where the graph of the equation crosses the x-axis.
At this point, the y-coordinate is always 0. Therefore, to find the x-intercept, you simply set y to 0 in the equation and solve for x.
For instance, in the equation given, \(x - 1.1y = 10\), we set \(y = 0\):

• Substitute 0 for y: \(x - 1.1 \times 0 = 10\)
• So, \(x = 10\)

This means the x-intercept is at (10, 0). Finding the x-intercept provides crucial information about the graph's behavior and location.

Exploring the Y-Intercept

The y-intercept plays a significant role in linear equations as it shows where the graph crosses the y-axis.
At this point, the x-coordinate is 0. To find the y-intercept, set x to 0 and solve for y.
In this situation, examining the equation \(x - 1.1y = 10\):

• Set \(x = 0\): \(0 - 1.1y = 10\)
• Solve for y: \(-1.1y = 10 \rightarrow y = -\frac{10}{1.1}\)

This results in the y-intercept (-\(\frac{10}{1.1}\), 0).The y-intercept provides insight into the graph's point of intersection with the y-axis and is often helpful in sketching the graph's outline.

Grasping Linear Equations

Linear equations are equations of the first degree, which means they involve variables raised only to the first power.
These equations typically take the form \(Ax + By = C\), where A, B, and C are constants.
A linear equation will produce a straight line when graphed. The x-intercept and y-intercept are critical in graphing linear equations because they offer two precise points that the line will pass through.
To graph such equations:

• Determine the x-intercept by setting y to 0
• Determine the y-intercept by setting x to 0
• Plot both points on their respective axes
• Draw a straight line through these points

Understanding these core features of linear equations makes it easier to interpret and graph them effectively.