Problem 12 - 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. $$4 x-5 y=10$$

Step-by-Step Solution

Answer

The x-intercept of the equation \(4x - 5y = 10\) is \(2.5\) and the y-intercept is \(-2\).

Step 1 - Find the x-intercept

Set \(y=0\) in the equation to find the x-intercept. The equation becomes \(4x - 5(0) = 10\), which simplifies to \(4x = 10\). Then, solve for \(x\) by dividing both sides of the equation by 4, so \(x = 10/4 = 2.5\). Therefore, the x-intercept of the equation is \(2.5\)

Step 2 - Find the y-intercept

Set \(x=0\) in the equation to find the y-intercept. The equation becomes \(4(0) - 5y = 10\), which simplifies to \(-5y = 10\). Then, solve for \(y\) by dividing both sides of the equation by -5, so \(y = 10 / -5 = -2\). Therefore, the y-intercept of the equation is \(-2\).

Key Concepts

Linear Equations

Linear equations are mathematical expressions that illustrate relationships between variables using straight lines on a graph. A linear equation typically takes the form of \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables.

This straightforward form allows us to calculate intercepts, which are key to understanding how linear equations behave on a graph. The intercepts are points where the line crosses the axes. Finding the intercepts can tell us a lot about the graph without needing to actually draw it.

In general terms:

• The **x-intercept** is where the graph crosses the x-axis. At this point, \( y \) is zero.
• The **y-intercept** is where the graph crosses the y-axis. At this point, \( x \) is zero.

Linear equations are foundational in algebra and have real-world applications like calculating costs, analyzing trends, and predicting outcomes.

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the values of variables that satisfy the equation. With linear equations, we often solve for one variable by isolating it on one side of the equation.

Let's explore how to solve for intercepts.

For the **x-intercept**:

• Set \( y = 0 \) and solve for \( x \).
• In our equation \( 4x - 5y = 10 \), setting \( y = 0 \) changes it to \( 4x = 10 \).
• Solving for \( x \) involves dividing both sides by 4, resulting in \( x = 2.5 \).

For the **y-intercept**:

• Set \( x = 0 \) and solve for \( y \).
• Substituting into the equation gives \( -5y = 10 \).
• Dividing both sides by \(-5\) solves the equation as \( y = -2 \).

Solving these equations helps determine crucial points on the graph, providing a deeper understanding of the equation's characteristics.

Algebra Concepts

Algebra uses symbols and letters to represent numbers and values in formulas and equations. This branch of mathematics is crucial in solving real-life problems and allows us to apply simple arithmetic rules to solve more complex questions.

Key algebra concepts present in solving intercepts include:

• **Substitution**: Replacing one part of an equation with its known value. This was done when we substituted \( y = 0 \) to find the x-intercept, and \( x = 0 \) to find the y-intercept.
• **Isolation**: Rearranging an equation to get the unknown variable by itself on one side. Solving for \( x \) and \( y \) involved isolating these variables through manipulation.
• **Balancing Equations**: Performing the same operations on both sides of an equation to maintain equality, essential for finding the correct intercepts.

These algebra concepts simplify seemingly complex problems and provide structured methods to finding solutions efficiently.

By understanding these foundational algebra ideas, you can tackle more extensive problems confidently.