Problem 127

Question

Describe a basic graphical method used to solve a linear equation. Give examples.

Step-by-Step Solution

Verified

Answer

Graph the equation and find where it intersects the x-axis.

1Step 1: Understanding the Linear Equation

A linear equation is an equation of the form $ax + b = 0$, where $a$ and $b$ are constants and $x$ is the variable. To solve this equation graphically, we will use the form $y = ax + b$. This allows us to plot it as a line on a graph.

2Step 2: Setting Up the Graph

To graph the equation $y = ax + b$, we need to set up a coordinate plane with an x-axis (horizontal) and a y-axis (vertical). We will plot points that satisfy the equation and then draw a line through these points.

3Step 3: Plotting Points for the Graph

Choose a value for $x$ and substitute it into the equation $y = ax + b$ to find the corresponding $y$ value. For example, pick $x = 0$, giving $y = b$ (the y-intercept). Choose a second $x$ value, say $x = 1$, which gives $y = a(1) + b$. Plot these points on the graph.

4Step 4: Drawing the Line

Once the points are plotted, draw a straight line through them. This line represents all solutions to the equation $y = ax + b$.

5Step 5: Finding the Solution

The solution to the original equation $ax + b = 0$ is the x-coordinate of the point where the line crosses the x-axis (i.e., where $y = 0$). Solve $0 = ax + b$ to find $x = -b/a$. Look at the graph to verify this x-coordinate as the point where the line intersects the x-axis.

Key Concepts

Linear EquationsCoordinate PlaneGraphing TechniquesY-intercept

Linear Equations

Linear equations are a fundamental part of mathematics that describe a straight line when graphed. The standard form of a linear equation is given as $ ax + b = 0 $, where $ a $ and $ b $ are constants and $ x $ is the variable. Another common form is $ y = ax + b $, which is useful for graphing because it shows the dependent variable $ y $ as a function of the independent variable $ x $. When you change the value of $ x $, the value of $ y $ changes linearly. Linear equations are called 'linear' because they graph as straight lines on a coordinate plane. These lines can slope upwards or downwards depending on the sign of $a$, or they can be horizontal, which occurs when $ a $ is zero, representing a constant function.

Coordinate Plane

A coordinate plane is a two-dimensional surface on which we can graph equations. It is divided by two perpendicular lines called axes. The horizontal line is known as the x-axis, and the vertical line is known as the y-axis. The point where these axes intersect is called the origin, which is designated by the coordinates $ (0,0) $. On this plane, every point can be identified by a pair of numerical coordinates $ (x, y) $, where $ x $ corresponds to the horizontal position and $ y $ corresponds to the vertical position of the point.
When plotting a linear equation like $ y = ax + b $, we use the coordinate plane to visualize the relationship between $ x $ and $ y $. This helps in understanding how changing $ x $ affects $ y $, and finding solutions to the equation becomes a matter of identifying points on this plane.

Graphing Techniques

Graphing techniques are essential for solving linear equations visually. To graph a linear equation like $ y = ax + b $, we start by selecting various values for $ x $ and computing the corresponding $ y $ values. These pairs $ (x, y) $ become points that we plot on the coordinate plane. A well-chosen set of points will provide a clear picture of the line.
Once you have two or more points, you draw a straight line through them. This technique is powerful because it visually represents all the solutions of the equation. The straight line indicates a constant rate of change, which is a key characteristic of linear equations.

Start by finding key points, like the y-intercept.
Choose different values for $ x $ to ensure an accurate graph.
Draw a straight line through plotted points, extending it across the plane.

These techniques provide a practical approach to understanding linear equations beyond just solving them algebraically.

Y-intercept

The y-intercept is a crucial concept in graphing linear equations. It is the point where the line crosses the y-axis. In the equation $ y = ax + b $, it occurs when $ x $ is zero. Substituting $ x = 0 $ into the equation gives $ y = b $. Therefore, $ b $ is the y-coordinate of the intercept, making the y-intercept the point $ (0, b) $.
This intercept is important because it gives us a starting point for graphing the line. Knowing where to begin simplifies plotting significantly. Moreover, it offers insight into the behavior of the linear equation without extensive calculations. The y-intercept in a graphing context: