Problem 47 - 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. $$ y=-6+3 x $$

Step-by-Step Solution

Answer

The x-intercept is (2,0) and the y-intercept is (0,-6).

Step 1: Calculate the X-Intercept

Set \(y = 0\) in the equation and solve for \(x\). The equation becomes: \(0 = -6 + 3x\). If you add 6 to each side and then divide each side by 3, you will find that \(x = 2\). So, the x-intercept of the line is (2,0).

Step 2: Calculate the Y-Intercept

Set \(x = 0\) in the equation and solve for \(y\). The equation becomes: \(y = -6 + 3 * 0\), which simplifies to \(y = -6\). So, the y-intercept of the line is (0,-6).

Step 3: Graph the Line and Label the Points

Plot the points (2,0) and (0,-6) on the graph. Draw a line through these points. Label the points where the line crosses the axes.

Key Concepts

linear equations

Linear equations are fundamental in algebra. They describe lines in a coordinate plane using a simple mathematical statement. A basic linear equation might look like this: \[ y = mx + b \] In this formula, \(m\) stands for the slope, indicating how steep the line is, and \(b\) is the y-intercept, which tells us where the line crosses the y-axis. Every straight line can be represented by such an equation.

• x and y: These are variables representing any point on the line.
• m: The slope, it tells us the direction and tilt of the line. If it's positive, the line ascends as it moves from left to right; if negative, it descends.
• b: The y-intercept, a constant that tells you where the line crosses the y-axis.

Linear equations are easy to solve and graph because of the straightforward nature of the line they describe.

graphing lines

Graphing lines on a coordinate plane allows you to visually comprehend linear equations. By plotting points on the axes, you can draw the line that matches the equation. Here's how you can do that practically: 1. **Find the intercepts**: These give you two points that the line passes through. As illustrated in the exercise, intercepts make it simpler to draft the line. 2. **Plot the intercepts**: For instance, with intercepts like (2,0) for the x-axis and (0,-6) for the y-axis, place these points on the graph. 3. **Draw the line**: Connect these points with a straight edge to form the line. Understanding how to graph lines connects algebra to geometry and allows you to interpret the meaning of equations visually on the plane.

coordinate plane

The coordinate plane is an essential tool in graphing linear equations. It consists of two perpendicular axes: - The **x-axis**, running horizontally. - The **y-axis**, running vertically. These axes divide the plane into four quadrants. On this grid, every point is assigned an ordered pair of numbers, like (x, y), which correspond to their positions relative to the two axes.

• Origin: Represents the point (0,0) where both axes intersect.
• Quadrants: Divided into four sections, with positive and negative values of x and y.

Working with the coordinate plane is crucial for linking algebraic calculations to geometric figures, helping you visualize and graph equations easily.

solving equations

Solving equations, like finding intercepts, involves basic algebraic manipulation to find unknown values. Let's break it down:1. **Identify the unknown**: What are you solving for? In finding intercepts, you're alternately solving for \(x\) and \(y\).2. **Substitute known values**: Set one variable to zero to find the intercept of the other. - For the x-intercept, set \(y = 0\), and solve for \(x\). - For the y-intercept, set \(x = 0\), and solve for \(y\).3. **Solve for the other variable**: Perform operations such as addition, subtraction, multiplication, or division to isolate the desired variable.Once solved, the intercepts can be plotted on the graph, making the conceptual connection between algebra and geometry clearer. Mastering solving equations is key to understanding deeper concepts in math.