Problem 20 - Algebra 1: Concepts and Skills

Question

Find the x-intercept of the line. $$ 2 x-3 y=6 $$

Step-by-Step Solution

Answer

The x-intercept of the line represented by the equation \(2x - 3y = 6\) is 3.

Step 1: Identify the Line Equation

The line equation given is \(2x - 3y = 6\). This is in the standard form for the equation of a line, which is \(Ax + By = C\). A, B, and C are constants. In our equation, A = 2, B = -3, and C = 6.

Step 2: Substitute y with 0

To find the x-intercept, the y value must be 0 because x-intercept is the point where the line crosses the x-axis. Substituting y in the equation results to \(2x - 3(0) = 6\), which simplifies to \(2x = 6\).

Step 3: Solve for x

Now, to get the x-intercept, the equation \(2x = 6\) must be solved for x. By dividing both sides of the equation by 2, the equation becomes \(x = 3\), which is the x-intercept of the line.

Key Concepts

Linear Equation

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations graph as straight lines because they have constant rates of change. They are simple yet powerful in describing various real-world situations where relationships exist between quantities. For example, a relationship between time and distance in motion problems.
The equation in the exercise, \(2x - 3y = 6\), is a linear equation because both x and y are variables raised to the first power, and they are combined with constants. Linear equations might sometimes appear in different forms, but they all depict a linear relationship.

Standard Form Equation

The standard form of a linear equation is written as \( Ax + By = C \), where A, B, and C are integers, and A should be a non-negative integer. The equation from the exercise, \(2x - 3y = 6\), fits this form with A = 2, B = -3, and C = 6.

• A is the coefficient of x, representing how changes in x affect the equation.
• B is the coefficient of y, representing how changes in y affect the equation.
• C is the constant term that affects the overall position of the line on the graph.

This form is particularly useful for solving equations and finding intercepts. It provides a clear way to identify coefficients for x and y, making the process of graphing or manipulation easier.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. In the context of linear equations, such as our given \(2x - 3y = 6\), solving can be straightforward. To find the x-intercept, we set y to zero because the x-intercept occurs where the line touches the x-axis, meaning no vertical displacement.
When y is zero, the equation simplifies to \(2x - 3(0) = 6\), which further simplifies to \(2x = 6\). This equation helps isolate x, making it easier to solve by performing arithmetic operations—specifically, division by the coefficient of x.

• First, simplify the equation if necessary.
• Then, isolate the variable by performing inverse operations, such as addition or division.
• Finally, check your solution by substituting it back into the original equation.

This method ensures the correctness of the obtained solution and adds clarity to the process of finding intercepts or points of interest in an equation.