Problem 7 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the \(x\) - and \(y\) -intercepts of the graph of the equation, if possible. $$y=x-5$$

Step-by-Step Solution

Answer

The x-intercept is \(x = 5\) and the y-intercept is \(y = -5\).

Step 1: Find the x-intercept

Set \(y = 0\) in the equation \(y = x - 5\). After setting \(y=0\), the equation becomes \(0 = x - 5\). Solving this equation gives the \(x\)-intercept, by adding \(5\) on both sides gives \(x = 5\).

Step 2: Find the y-intercept

Set \(x = 0\) in the equation \(y = x - 5\). After setting \(x=0\), the equation becomes \(y = 0 - 5\) which simplifies to \(y = -5\). So, the \(y\)-intercept is \(y=-5\).

Key Concepts

Graphing Linear Equations

Understanding how to graph linear equations is a fundamental skill in algebra that allows students to visualize problems and solutions. A linear equation can be represented on a coordinate plane—where the 'x' and 'y' axes intersect is known as the origin. Each point on this graph corresponds to a possible solution to the equation.

When graphing the equation such as \(y = x - 5\), you would start by plotting the intercepts. These are the points where the line crosses the \(x\)-axis and \(y\)-axis, which are also the keys to drawing the line. The \(x-\) and \(y-\)intercepts often provide the easiest points to find and plot. To graph the equation, simply draw a straight line through these intercepts—since it's a linear equation, only two points are necessary to determine the line.

Algebraic Solutions

Solving algebraic expressions and equations is pivotal for understanding complex mathematical concepts. It involves finding the values for the unknown variables that make the equation true. The step-by-step example in the exercise illustrates the algebraic process to find the \(x\)- and \(y\)-intercepts of a given equation.

An \(x-\)intercept is found by setting \(y=0\) and solving the resulting equation for \(x\), which denotes where the graph crosses the \(x\)-axis. Conversely, a \(y-\)intercept is found by setting \(x=0\) and solving for \(y\), representing where the graph crosses the \(y\)-axis. These calculated intercepts (\(x = 5\) and \(y = -5\) in our example) serve as crucial checkpoints in the graph and as foundational elements in understanding the structure of algebraic solutions.

Slope-Intercept Form

The slope-intercept form, presented as \(y = mx + b\), is a quick method to write the equation of a straight line. In this format, 'm' stands for the slope of the line, which expresses the steepness and direction, and 'b' represents the \(y\)-intercept, the point where the line crosses the vertical \(y\)-axis.

In the exercise \(y = x - 5\), the slope \(m\) is 1 (as implied by the coefficient of \(x\), since it's not explicitly written), and the \(y\)-intercept \(b\) is -5. Using the slope-intercept form makes both graphing and interpreting linear equations more intuitive. For instance, it immediately reveals that for every one unit increase in \(x\), the value of \(y\) increases by the same amount due to the slope of 1. This form is particularly useful because it highlights the two fundamental features of linear equations—the slope and the \(y\)-intercept—directly.