Problem 24 - Algebra 1: Concepts and Skills

Question

Find the x-intercept of the line. $$ -x-5 y=12 $$

Step-by-Step Solution

Answer

The x-intercept of the line is -12.

Step 1: Setting up the problem

Write down the given equation, which is \(-x-5y=12\)

Step 2: Calculating the x-intercept

Set y equal to 0 in the equation. The equation becomes \(-x-5(0)=12, or -x=12\)

Step 3: Solving for x

We solve for x by multiplying both sides of the equation by -1. That results in \(x=-12\)

Key Concepts

Linear Equations

Linear equations are mathematical expressions that form straight lines when graphed on a coordinate plane. They typically have variables that are raised only to the power of one, which is why they are called 'linear.' A standard linear equation in two variables is often written in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants.

Understanding the structure of linear equations is vital because it helps us identify components like the slope and intercepts. The equation provided in our exercise, \(-x - 5y = 12\), is a linear equation, meaning it demonstrates a constant rate of change between variables \(x\) and \(y\).

In a linear equation, both x and y can take virtually any value that satisfies the equation, resulting in a line with an infinite number of solutions. These solutions are the points through which the line passes, showing the relationship between \(x\) and \(y\).

Solving for x

Solving for x means finding the value of x that satisfies the equation, especially when given certain conditions. In the context of finding the x-intercept of a line, we need to set \(y = 0\) because the x-intercept is the point where the line crosses the x-axis.

Let's take the equation \(-x - 5y = 12\) from our exercise. By replacing \(y\) with 0, we transform the equation to \(-x - 5(0) = 12\), simplifying to \(-x = 12\).

To isolate \(x\), we multiply both sides of the equation by \(-1\), giving us \(x = -12\). This method of manipulating equations helps us find the specific solution for \(x\), demonstrating that at the point \(-12, 0\), the line intersects the x-axis.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometric figures through algebraic equations. It uses coordinate points to represent geometric shapes and analyze their properties.

The x-intercept found in our exercise, \((-12, 0)\), is a classic example of how coordinate geometry relates algebra and geometry. On a Cartesian plane, the x-intercept is the point that lies on the x-axis where \(y\) equals zero. This point tells us where a line crosses the x-axis.

• The x-axis runs left to right (horizontally) and serves as the reference line for measuring the x-value.
• The y-axis runs up and down (vertically) and measures the y-value.

By understanding the x-intercepts, we gain insights into the behavior of linear functions and how they interact with the axes, which is a fundamental aspect of coordinate geometry.