Problem 45 - Algebra 1

Question

Find the \(x\) -intercept and the \(y\) -intercept of the line. Graph the equation. Label the points where the line crosses the axes. $$ y=x-3 $$

Step-by-Step Solution

Answer

The \(x\)-intercept of the line \(y = x - 3\) is \(x = 3\), and the \(y\)-intercept is \(y = -3\).

Step 1: Find the x-intercept

To find the \(x\)-intercept, set \(y = 0\) in the equation and solve for \(x\). Doing so gives the equation \(0 = x - 3\), which simplifies to \(x = 3\). \nThus, the \(x\)-intercept is 3.

Step 2: Find the y-intercept

To find the \(y\)-intercept, set \(x = 0\) in the equation and solve for \(y\). This leads to the equation \(y = 0 - 3\), which simplifies to \(y = -3\). \nThus, the \(y\)-intercept is -3.

Step 3: Graph the equation and label the intercepts

Sketch a straight line that intercepts the \(x\)-axis at \(x = 3\) and the \(y\)-axis at \(y = -3\). Label these points properly. Remember that the \(x\)-intercept is the point (3, 0) and the \(y\)-intercept is the point (0, -3).

Key Concepts

X-Intercept

Understanding the x-intercept is crucial when graphing linear equations. It's the point where the line crosses the x-axis. To find it, set the value of y to zero in the equation of the line, as the x-axis corresponds to where y is always zero.
In our exercise, with the equation \(y = x - 3\), setting y to zero gives us \(0 = x - 3\). Solving for x reveals that \(x = 3\), which means that the x-intercept is at the point (3, 0). This tells us that the line touches the x-axis exactly three units to the right of the origin.

Y-Intercept

The y-intercept, similarly, is where the line crosses the y-axis, and it's found by setting the x-value to zero. This is because the y-axis is defined by the locations where x is zero.
In the equation \(y = x - 3\), when we set \(x = 0\), the equation simplifies to \(y = -3\). Thus, the y-intercept is at the point (0, -3), meaning the line crosses the y-axis three units below the origin. Identifying the y-intercept is fundamental in understanding how a linear equation behaves and is a starting point in creating its graph.

Algebraic Solutions

To graph a line properly, we often use algebraic solutions to understand the equation's behavior. By finding the x and y intercepts, as shown in the example, we gather specific points which we can then plot. These algebraic manipulations simplify the equation to where it only includes one unknown, either x or y, allowing us to solve for that variable with ease.
Algebra offers a powerful toolkit to deconstruct and solve equations systematically, which paves the way for accurate graphing and assists us in predicting how changes in the equation will affect its graph.

Coordinate Graphing

Once we have the intercepts from our algebraic solutions, we can use coordinate graphing to visualize the equation. Sketching a graph involves plotting points on a coordinate plane where each point represents a solution – a pair of x and y values – to the equation.
For the linear equation \(y = x - 3\), after plotting the x-intercept (3, 0) and y-intercept (0, -3), we draw a straight line through these two points, extending it to cover the plane. This visual representation is an effective way to see the relationship between x and y and how they coincide with solutions to the equation.