Problem 55 - Algebra 1: Concepts and Skills

Question

Find the \(x\) -intercept of the graph of the equation \(3 x+y=-9\) A. \(-3\) B. 3 C. 9 D. -9

Step-by-Step Solution

Answer

The x-intercept is A. \(-3\)

Step 1: Substitute y = 0 into the Equation

The x-intercept is the x-value when y = 0. So, substitute y = 0 into the equation \(3x + y = -9\). This gives \(3x + 0 = -9\), which simplifies to \(3x = -9\).

Step 2: Solve for x

Divide each side of the equation by 3 to solve for x. This gives \(x = -9 / 3\).

Step 3: Simplify the Result

Simplify \(x = -9 / 3\) to get \(x = -3\)

Key Concepts

Understanding Linear Equations

Linear equations are the simplest form of equations we encounter in algebra. They have the general form of ax + by = c, where a, b, and c are constants and x and y are variables. The solutions to these equations are ordered pairs that satisfy both variables.

Let's take, for example, the linear equation from the exercise, 3x + y = -9. The constants here are 3 and -9, the variable x is multiplied by 3, and y is not multiplied by any number other than 1 (which is typically omitted). Equations like these are solved by isolating one variable and finding its value in terms of the other or in terms of the constants.

Graphing Linear Equations

To graph a linear equation, such as 3x + y = -9, we need at least two points to draw the line. One way to find a point is by identifying the x-intercept, where the line crosses the x-axis, and similarly, the y-intercept, where the line crosses the y-axis.

In graphical terms, the x-intercept occurs when y = 0, simplifying the equation to solve only for x. Once the x-intercept is found, it serves as a pivotal point in plotting the line and understanding the graph's orientation. This concept is fundamental since it also helps in understanding the slope and direction of the line.

Solving for x

To solve for x in a linear equation, we aim to isolate x on one side of the equation. Doing so generally involves two steps: rearranging the equation and performing arithmetic operations.

Considering the equation from our example, 3x + y = -9, we first identify what value of x will make the equation true when y is zero, which leads us to 3x + 0 = -9. We then divide both sides of the equation by the coefficient of x, which is 3, to isolate x. This leaves us with x = -3, which is the x-intercept of the equation's graph.

Understanding how to isolate and solve for x is not only critical for finding intercepts but is also a core skill in algebra used to solve various types of problems.