Problem 17 - Introductory and Intermediate Algebra for College Students 4th

Question

Find the \(x\) -intercept and the \(y\) -intercept of the graph of each equation. Do not graph the equation. $$2 x=3 y-11$$

Step-by-Step Solution

Answer

The \(x\)-intercept is \(-5.5\) and the \(y\)-intercept is \(\frac{11}{3}\)

Step 1: Find the \(x\)-intercept

Set \(y=0\) in the equation and solve for \(x\).\nFrom \(2x=3(0)-11\), we get \(x=-5.5\) .

Step 2: Find the \(y\)-intercept

Set \(x=0\) in the equation and solve for \(y\).\nFrom \(2(0)=3y-11\), we get \(y=\frac{11}{3} \).

Key Concepts

X-Intercept

The x-intercept of a graph refers to the point where the graph crosses the x-axis. It is the location on the graph where the value of the y-coordinate is zero. To find this point in the context of algebraic equations, one simply sets the y-variable to zero and then solves the resulting equation for x.
This method works because when y is zero, the point is located on the x-axis itself. In our exercise, we applied this method to the equation 2x = 3y - 11. By setting y to zero, the equation becomes 2x = -11, which simplifies to x = -5.5 when solved. Therefore, the x-intercept is at the point (-5.5, 0).

Remembering the x-intercept

• It is found where the graph crosses the x-axis.
• Always set y to zero in the equation.
• The resulting x-value gives the x-intercept.

Y-Intercept

Conversely, the y-intercept is where the graph intersects with the y-axis, hence the name. This intersection point arises where the x-coordinate is zero. To find the y-intercept algebraically, set the x-variable to zero and solve for y in the equation.
In the example equation provided, 2x = 3y - 11, when we assign x the value of zero, the equation becomes 3y = 11, which yields y = 11/3 when solved. Thus, the y-intercept is (0, 11/3), showing the exact point where the line would intersect the y-axis on a graph.

Key Points for the y-intercept

• Located where the line crosses the y-axis.
• x is set to zero to find the y-intercept.
• The y-intercept is expressed as a point with an x-coordinate of zero.

Algebraic Equations

An algebraic equation consists of variables, coefficients, and constants that are connected with arithmetic operations and equated to something, often zero or another expression. Algebraic equations can represent lines, curves, and various other relationships between quantities.
These equations are the foundation of algebra and are used to solve numerous problems by finding the values of the unknown variables. The example provided in the exercise, 2x = 3y - 11, is an algebraic equation which we manipulate to find the x-intercept and y-intercept without graphing it.

Components of Algebraic Equations

• Variables: Letters representing unknowns (e.g., x and y).
• Coefficients: Numbers multiplying the variables.
• Constants: Numbers on their own, not multiplying a variable.

Algebraic equations are crucial in determining the relationships of variables and are the initial step in graphing functions or interpreting their graphs.

Linear Equations

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 produce straight lines when graphed on a coordinate plane. The general form of a linear equation in two variables x and y is usually written as ax + by = c, where a, b, and c are constants.
These are called linear because the graph of these equations reveals a straight line. They are fundamental in understanding how changes in one variable reflect changes in another, which is the concept of slope — the rate of change. For our exercise equation, 2x = 3y - 11, rearranging it provides us with a linear equation format: 3y = 2x + 11 or in the 'y = mx + b' format, y = (2/3)x + 11/3, where the slope (m) is 2/3 and the y-intercept (b) is 11/3.

Linear Equation Characteristics

• Form straight lines when graphed.
• They have a constant rate of change or slope.
• Simplest to use when finding intercepts algebraically.