Problem 26 - Algebra 1: Concepts and Skills

Question

Find the x-intercept of the line. $$ -13 x-y=39 $$

Step-by-Step Solution

Answer

The x-intercept of the line -13x - y = 39 is -3.

Step 1: Rearrange the Equation

In the first step, take the equation -13x - y = 39 and rearrange it to isolate the y term. Do this by adding 13x to both sides of the equation which results in -y = 13x + 39.

Step 2: Set y to Zero

Since the x-intercept of a line is the point where the line crosses the x-axis, where y = 0, set y to 0 and solve the equation for x. This results in: 0 = 13x + 39.

Step 3: Solve for x

Finally, solve the equation 0 = 13x + 39 for x. To do this, subtract 39 from both sides, which gives -39 = 13x. Then, divide both sides of the equation by 13 to solve for x which results in \( x = -3.

Key Concepts

Linear Equation

A linear equation is an algebraic expression that represents a straight line when graphed on a coordinate plane. It involves variables with only the first power, meaning no exponents greater than one are present. The general form of a linear equation in two variables is \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants. In our example, we have the equation \(-13x - y = 39\).

This type of equation is fundamental in algebra due to its simplicity and utility in modeling real-world situations.

• Linear equations can describe relationships where one quantity depends directly on another.
• They are found in different formats, including slope-intercept form \(y = mx + b\), which makes it easier to identify the slope \(m\) and y-intercept \(b\).

Understanding how to work with linear equations, such as rearranging them into different forms, is essential when analyzing linear relationships. Let's further explore how these equations interact with the coordinate plane.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional plane consisting of an x-axis and a y-axis that intersect at the origin (0, 0). It allows us to visually represent algebraic equations, making it easier to understand the relationships between variables.

When graphing a linear equation like \(-13x - y = 39\), you plot points that satisfy the equation:

• The intersection of this line with the x-axis is the x-intercept, where \(y = 0\).
• The y-intercept is the point where the line crosses the y-axis, \(x = 0\).

By determining these intercepts, you can easily draw the graph of the equation.

Our goal was to find the x-intercept of the given equation. After rearranging and solving it, we found the intersection point \((-3, 0)\). This point signifies where the line touches the x-axis, providing a visual understanding of the equation's relationship on a coordinate plane.

Solving Equations

Solving equations is the process of finding the unknown variable that makes the equation true. In the context of finding the x-intercept, we set \(y=0\) and solved for \(x\).

Let's break down the steps used in our example:

• We started with the equation \(-13x - y = 39\) and rearranged it by isolating terms with \(y\).
• Since we're interested in the x-intercept, we set \(y = 0\), transforming the equation into \(0 = 13x + 39\).
• By subtracting 39 from both sides, we obtained \(-39 = 13x\).
• Finally, dividing by 13, we found \(x = -3\).

This straightforward process highlights a key aspect of solving linear equations: systematically isolating the variable to find the solution.

Knowing how to solve such equations is crucial for interpreting graphs, predicting outcomes, and understanding mathematical concepts in both academic and practical contexts.