Problem 16 - Introductory and Intermediate Algebra for College Students 4th

Question

Find the \(x\) -intercept and the \(y\) -intercept of the graph of each equation. Do not graph the equation. $$8 x-11 y=0$$

Step-by-Step Solution

Answer

The \( x \) -intercept and the \( y \) -intercept of the graph of the equation are both 0.

Step 1: Find the \( x \) intercept

To find the \( x \) -intercept, set \( y = 0 \) in the equation and solve for \( x \). So from the given equation \(8x - 11y = 0 \), when \( y = 0\), the equation becomes \(8x = 0 \). Solving for \( x \), we get \( x = 0 \)

Step 2: Find the \( y \) intercept

To find the \( y \) -intercept, set \( x = 0 \) in the equation and solve for \( y \). So from the equation \( 8x - 11y = 0 \), when \( x = 0 \), the equation becomes \( -11y = 0 \). Solving for \( y \), we get \( y = 0 \)

Key Concepts

Equation Solving

Equation solving is fundamental in algebra, and it involves finding the value or values for variables that make an equation true. In an equation like \(8x - 11y = 0\), the objective is to manipulate the equation to find specific variable values.
To effectively solve equations, we often isolate a variable by using arithmetic operations such as addition, subtraction, multiplication, or division. For this linear equation, solving for one variable is accomplished by substituting a specific value for the other variable (like setting \(y = 0\) to find the \(x\)-intercept).
This process of substitution helps break down the problem, making it manageable and straightforward.

X-Intercept

The \(x\)-intercept of a line is the point where the line crosses the \(x\)-axis. To find the \(x\)-intercept, set \(y = 0\) in the equation and solve for \(x\). This is because any point on the \(x\)-axis has a \(y\)-value of 0.
In our exercise, with the equation \(8x - 11y = 0\), setting \(y = 0\) gives \(8x = 0\). Solving for \(x\), we find that \(x = 0\). Thus, the \(x\)-intercept of the line is at the point (0, 0), meaning the line crosses the \(x\)-axis at the origin.

Y-Intercept

The \(y\)-intercept of a line is where the line crosses the \(y\)-axis. To find this intercept, set \(x = 0\) and solve for \(y\). That's because any point on the \(y\)-axis has an \(x\)-value of 0.
For the equation \(8x - 11y = 0\), setting \(x = 0\) transforms it into \(-11y = 0\). Solving for \(y\), we discover that \(y = 0\). This means the \(y\)-intercept is also at the origin (0, 0).
In this particular example, both the \(x\) and \(y\)-intercepts coincide at the point (0, 0).

Linear Equations

Linear equations, such as \(8x - 11y = 0\), are equations involving two variables and are plotted as straight lines on a graph. These lines can be defined in slope-intercept form, as \(y = mx + c\), where \(m\) indicates the slope and \(c\) the \(y\)-intercept, or in standard form \(Ax + By = C\).
This equation is already in its simplest form but does not have a traditional slope-intercept form because it passes through the origin. Here, both coefficients \(A = 8\) and \(B = -11\) determine how steep the line is and in which direction it leans.
Linear equations are foundational because they model so many real-world situations, from calculating expenses to predicting growth.