Problem 5 - College Algebra

Question

For each of the following exercises, find the \(x\)-intercept and the \(y\)-intercept without graphing. Write the coordinates of each intercept. $$y=-3 x+6$$

Step-by-Step Solution

Answer

The x-intercept is (2, 0) and the y-intercept is (0, 6).

Step 1: Identifying the Equation

The given equation is in the slope-intercept form, which is \(y = mx + b\). Here, \(m = -3\) and \(b = 6\).

Step 2: Finding the x-Intercept

To find the \(x\)-intercept, set \(y = 0\). Substitute \(0\) for \(y\) in the equation: \(0 = -3x + 6\).

Step 3: Solving for x in x-Intercept

Solve \(0 = -3x + 6\) for \(x\):\[ -3x + 6 = 0 \rightarrow -3x = -6 \rightarrow x = 2 \]. So the \(x\)-intercept is \((2, 0)\).

Step 4: Finding the y-Intercept

To find the \(y\)-intercept, set \(x = 0\). Substitute \(0\) for \(x\) in the equation: \(y = -3(0) + 6\).

Step 5: Solving for y in y-Intercept

Solve \(y = -3(0) + 6\): \[ y = 0 + 6 \rightarrow y = 6 \]. So the \(y\)-intercept is \((0, 6)\).

Key Concepts

Slope-Intercept Form

A linear equation describes a straight line when plotted on a graph. One of the most common ways to express this equation is in the slope-intercept form. In this form, the equation of a line is represented as \(y = mx + b\). Here, \(m\) stands for the slope of the line, while \(b\) indicates the \(y\)-intercept. This format makes it incredibly easy to determine how steep the line is and where it crosses the \(y\)-axis.

In our given problem, the equation is \(y = -3x + 6\). Breaking it down, we see that the slope \(m\) is \(-3\), meaning the line falls three units vertically for every unit it moves horizontally.
The \(y\)-intercept \(b\) is \(6\), which is where the line crosses the \(y\)-axis when \(x=0\).
Knowing the slope and intercept can help us quickly sketch out the basic characteristics of the line without graphing it.

Linear Equations

Linear equations are algebraic expressions that demonstrate a clear relationship between two variables, usually \(x\) and \(y\). These variables are not raised to any power higher than one, ensuring the graph's shape remains a line.

Given, the equation \(y = -3x + 6\) is a linear equation. It comprises two main elements:

• The slope: Is the coefficient of \(x\), which defines the steepness and direction of the line. Here, it is \(-3\).
• The y-intercept: It indicates where the line crosses the \(y\)-axis. In this equation, it's \(6\).

Linear equations are straightforward to manage and are foundational to understanding more complex equations. Appreciating the structure allows quick determinations of intercepts, which are key points on the line.

Solving Equations

To find the intercepts of a linear equation, we solve it to find specific values. For the \(x\)-intercept, set \(y=0\) and solve for \(x\). For the \(y\)-intercept, set \(x=0\) and solve for \(y\).

Let's break it down with our example:

• Finding the \(x\)-Intercept: Start by setting \(y=0\) in \(y = -3x + 6\):
  \[0 = -3x + 6\] Rearrange to solve for \(x\):
  \[-3x = -6\]
  Divide both sides by \(-3\):
  \[x = 2\]
  Thus, the \(x\)-intercept is \((2, 0)\).


• Finding the \(y\)-Intercept: Set \(x=0\) in the equation:
  \[y = -3(0) + 6\]
  Simplify:
  \[y = 6\]
  Therefore, the \(y\)-intercept is \((0, 6)\).

By following these steps, we can easily find intercepts which are crucial for graphing linear equations and understanding their behavior.