Problem 5 - College Algebra with Corequisite Support

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: Understanding the Intercepts

The intercepts are the points where the graph of a line crosses the axes. The \(x\)-intercept occurs where \(y = 0\) and the \(y\)-intercept occurs where \(x = 0\).

Step 2: Finding the x-intercept

To find the \(x\)-intercept, we set \(y = 0\) in the equation. Start with the equation: \(y = -3x + 6\).Set \(y = 0\):\(0 = -3x + 6\).Solve for \(x\) by adding 3\(x\) on both sides: \(3x = 6\).Divide both sides by 3: \(x = 2\).So, the \(x\)-intercept is \((2, 0)\).

Step 3: Finding the y-intercept

To find the \(y\)-intercept, we set \(x = 0\) in the equation.Use the equation \(y = -3x + 6\).Set \(x = 0\):\(y = -3(0) + 6\).Calculate \(y\): \(y = 6\).The \(y\)-intercept is \((0, 6)\).

Key Concepts

finding x-intercept

Finding the x-intercept of a linear equation involves determining the point where the line crosses the x-axis. Since the line intersects the x-axis, the value of y is zero at this point. We start by having an equation in the form of:

• y = mx + b, where m is the slope and b is the y-intercept.

Inputting y = 0 into the equation allows us to solve for x. For example, if we have the equation y = -3x + 6, set 0 = -3x + 6 and solve for x:

• Add 3x to both sides, getting 3x = 6.
• Then, divide both sides by 3 to find x = 2.

The x-intercept, therefore, is the point (2, 0), indicating the line crosses the x-axis at x = 2.

finding y-intercept

The y-intercept is found where the graph of the equation crosses the y-axis. This is where the value of x is zero. Substituting zero for x in the given equation allows us to find y. Take y = -3x + 6, for instance:

• Set x = 0: The equation becomes y = -3(0) + 6.
• Calculate the result, which simplifies to y = 6.

The y-intercept is, thus, at the point (0, 6). This tells us the line crosses the y-axis at y = 6.

linear equations

Linear equations are mathematical expressions that establish a relationship between variables, producing a straight line when graphed. The general form of a linear equation is:

• y = mx + b

where y is the dependent variable, x is the independent variable, m represents the slope of the line, and b signifies the y-intercept. The slope m indicates how steep the line is. If the slope is positive, the line rises as it moves from left to right. Conversely, a negative slope means the line falls. The y-intercept b is the point where the line crosses the y-axis, and it indicates the value of y when x is zero. Understanding these elements is crucial, as they help us graph the equation and understand how variables relate within it. Linear equations can model real-world scenarios such as calculating costs or predicting trends.