Problem 55 - Algebra 2

Question

What are the intercepts of \(-3 x+5 y-2 z=60 ?\) $$ \begin{array}{ll}{\text { A. } x=-180, y=300, z=-120} & {\text { B. } x=-20, y=12, z=-30} \\ {\text { C. } x=-3, y=5, z=-2} & {\text { D. } x=-60, y=60, z=-60}\end{array} $$

Step-by-Step Solution

Answer

The intercepts are x = -20, y = 12, z = -30. The correct answer is B.

Step 1: Find the x-intercept

To find the x-intercept, set y and z to zero and solve the equation for x. So, the equation becomes \( -3x = 60 \). Divide both sides by -3 to get the x-intercept.

Step 2: Find the y-intercept

To find the y-intercept, set x and z to zero and solve the equation for y. So, the equation becomes \( 5y = 60 \). Divide both sides by 5 to get the y-intercept.

Step 3: Find the z-intercept

To find the z-intercept, set x and y to zero and solve the equation for z. So, the equation becomes \( -2z = 60 \). Divide both sides by -2 to get the z-intercept.

Key Concepts

X-intercept

Understanding the x-intercept of a linear equation is crucial to comprehending how a line crosses the x-axis on a coordinate plane. The x-intercept occurs where the value of the y and any other variable, if present, like z, are set to zero because these points lie on the x-axis itself.

To find the x-intercept from an equation like \( -3 x+5 y-2 z=60 \), follow the initial step by setting \( y \) and \( z \) to zero, simplifying the equation to \( -3x = 60 \). Solving for \( x \) becomes straightforward by dividing both sides by -3, yielding \( x = -20 \). This single point \( (-20, 0, 0) \) is where the linear equation crosses the x-axis.

Y-intercept

Moving on to the y-intercept, this is the point at which the line passes through the y-axis. At this point, the values of all other variables, like x and z in our three-dimensional example, are zero.

When we look at our equation \( -3 x+5 y-2 z=60 \) and set \( x \) and \( z \) to zero, we are left with \( 5y = 60 \). By dividing both sides by 5, we can easily determine the y-intercept to be \( y = 12 \). The coordinate corresponding to the y-intercept is \( (0, 12, 0) \) which shows where the line meets the y-axis.

Z-intercept

The concept of the z-intercept becomes relevant in three-dimensional geometry, where we extend the idea of lines crossing axes to the z-axis. Similarly to finding the x-intercept and y-intercept, to find the z-intercept, set the other variables to zero— in this case, x and y.

Using our given equation, we have \( -2z = 60 \) after setting \( x = 0 \) and \( y = 0 \). After dividing both sides by -2, we find that \( z = -30 \), marking the point \( (0, 0, -30) \) as the z-intercept. This point signifies where the line intersects the z-axis in a three-dimensional space.

Solving Linear Equations

Solving linear equations, whether they are in two or three-dimensional space, is a foundational skill in algebra. To solve these equations, one must isolate the variable of interest on one side of the equation by using arithmetic operations like addition, subtraction, multiplication, or division.

In the context of finding intercepts, variables not of immediate interest are set to zero to find the specific 'intercept' point on that axis. This step-by-step approach simplifies the equation to one variable, enabling you to solve for that lone variable, as demonstrated in the earlier steps. It's important to remember to perform the same operations on both sides of the equation to maintain its balance and arrive at the correct solution.