Problem 44 - College Algebra with Corequisite Support

Question

For the following exercises, find the \(x\) - and \(y\) -intercepts of each equation. $$ k(x)=-5 x+1 $$

Step-by-Step Solution

Answer

The \(y\)-intercept is \((0, 1)\) and the \(x\)-intercept is \(\left(\frac{1}{5}, 0\right)\).

Step 1: Identify the Equation of the Line

The provided equation of the line is \(k(x) = -5x + 1\). Our task is to find the \(x\)- and \(y\)-intercepts.

Step 2: Find the y-Intercept

The \(y\)-intercept is the value of the function when \(x = 0\). Substitute \(x = 0\) into the equation: \(k(0) = -5(0) + 1 = 1\). Thus, the \(y\)-intercept is \((0, 1)\).

Step 3: Find the x-Intercept

The \(x\)-intercept is the value of \(x\) when \(k(x) = 0\). Set the equation \(-5x + 1 = 0\) and solve for \(x\): \(-5x = -1\) which simplifies to \(x = \frac{1}{5}\). Thus, the \(x\)-intercept is \(\left(\frac{1}{5}, 0\right)\).

Key Concepts

Linear Equations

Linear equations are mathematical expressions of the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. These equations represent straight lines when graphed on a coordinate plane. Each unique linear equation describes a particular line, characterized by these two constants.

• The slope \( m \) indicates the steepness and direction of the line. A positive slope means the line rises as it moves from left to right, while a negative slope means it falls.
• The y-intercept \( b \) is where the line crosses the y-axis, giving a point with coordinates \((0, b)\).

Linear equations are fundamental in algebra, forming the basis for solving complex mathematical models and systems.

Intercepts

Intercepts are points where a graph intersects the axes of a coordinate plane. They are crucial in understanding and visualizing equations.

• The **y-intercept** is where the graph crosses the y-axis. To find it, set \( x = 0 \) in the equation and solve for \( y \). In our example, we found the y-intercept is at \((0, 1)\).
• The **x-intercept** is where the graph crosses the x-axis. To determine this, set the entire expression equal to zero and solve for \( x \). In our linear function, the x-intercept is \(\left(\frac{1}{5}, 0\right)\).

In practical terms, intercepts can represent scenarios like where a company’s revenue might equal its costs, making profits or losses zero.

Solving Equations

Solving equations is a skill used to find unknown values. It involves manipulating an equation to isolate the variable of interest. Here, we'll focus on how to find intercepts.

• **To find an x-intercept**: The process involves setting the equation value to zero. For \( -5x + 1 = 0 \), simplify to find \( x = \frac{1}{5} \).
• **For a y-intercept**: Simply substitute \( x = 0 \) into the equation and solve for \( y \), giving you the intercept directly.

The steps in equation-solving often include adding, subtracting, multiplying, or dividing both sides of the equation to keep it balanced while isolating variables. This core strategy is essential, facilitating a deeper understanding of algebraic principles.