Problem 17 - Introductory Algebra for College Students

Question

Find the \(x\)-intercept and the \(y\)-intercept of the graph of each equation. Do not graph the equation. \(2 x=3 y-11\)

Step-by-Step Solution

Answer

The y-intercept of the line is \(y = 11/3\) and the x-intercept of the line is \(x = -11/2\).

Step 1: Find the Y-intercept

The \(y\)-intercept is found by setting \(x\) to 0 in the equation. So, replace \(x\) with 0 in the equation \(2x=3y-11\) to get \(0=3y-11\). Now, solve this for \(y\) by adding 11 to both sides to get \(3y = 11\), and then dividing both sides by 3 to solve for \(y\), giving \(y=11/3\).

Step 2: Find the X-intercept

The \(x\)-intercept is found by setting \(y\) to 0 in the equation. So, replace \(y\) with 0 in the equation \(2x=3y-11\) to get \(2x=3*0-11\) or \(2x=-11\). Now, solve this for \(x\) by dividing both sides by 2 to get \(x=-11/2\).

Key Concepts

Linear Equations in Algebra

When entering the world of algebra, you'll quickly encounter linear equations. These are equations where variables are raised to the power of one and are plotted as straight lines on a graph. A classic form of a linear equation is given by the slope-intercept form, which looks like:
\( y = mx + b \)

• \( m \) represents the slope, which indicates the steepness of the line.
• \( b \) is the y-intercept, where the line crosses the y-axis.

The equation from the exercise, \( 2x = 3y - 11 \), can be rearranged to this form with some basic manipulation. This involves isolating \( y \) on one side to see how it's changes affect the graph. Understanding how to manipulate and graph these equations is crucial as they form the foundation for more complex algebraic explorations.

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the value(s) of the variable(s) that make the equation true. It's like solving a puzzle where you need to figure out the missing pieces. The key steps include:

• Combining like terms.
• Using the additive and multiplicative properties of equality to isolate the variable.

In our example, to find the intercepts, we essentially solve two separate equations, one for when \( x = 0 \) and one for when \( y = 0 \), simplifying each to discover the points where the graph crosses the axes. By mastering these techniques, you'll be equipped to tackle a variety of problems in algebra.

Intercepts of a Graph

Intercepts of a graph tell us where a function or equation crosses the axes of the coordinate system. Specifically, the y-intercept is where the graph crosses the y-axis, and the x-intercept is where it crosses the x-axis. Here's how we find them:

• To find the y-intercept, set \( x = 0 \) in the equation and solve for \( y \).
• To find the x-intercept, set \( y = 0 \) in the equation and solve for \( x \).

These points are often used to quickly sketch the graph of the equation without plotting multiple points and can provide insights into the behavior of the equation. In the given exercise, understanding intercepts helps to identify key characteristics of the graph without actually drawing it.

Algebraic Methods

Algebraic methods refer to the set of techniques used to manipulate and solve equations. These can include:

• Distributive property to remove parentheses.
• Addition and subtraction to move terms from one side of an equation to the other.
• Multiplication and division to isolate the variable.
• Factoring to simplify expressions and solve quadratic equations.

Applying the correct algebraic methods allows you to simplify the original equation into a more manageable form, thus making it easier to find the solution. For example, to find the intercepts in our exercise, we rearrange and isolate variables appropriately, demonstrating how algebraic methods facilitate the solving process.