Problem 33 - Elementary Algebra

Question

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing. $$ \left\\{\begin{array}{l} x-3 y=-3 \\ y=2 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \( (3, 2) \).

Step 1: Write each equation in slope-intercept form

The equations are: 1) \( x - 3y = -3 \) 2) \( y = 2 \)The second equation is already in slope-intercept form (\( y = mx + b \)) as \( y = 2 \). For the first equation, solve for y:\[ x - 3y = -3 \]Subtract \( x \) from both sides:\[ -3y = -x - 3 \]Divide by \( -3 \):\[ y = \frac{-1}{-3}x + \frac{-3}{-3} \]\[ y = \frac{1}{3}x + 1 \]

Step 2: Identify the y-intercepts and slopes

For \( y = \frac{1}{3}x + 1 \), the y-intercept is 1 and the slope is \( \frac{1}{3} \).For \( y = 2 \), the y-intercept is 2 and the slope is 0.

Step 3: Graph the equations

Graph \( y = \frac{1}{3}x + 1 \): - Start at the y-intercept (0, 1). - Use the slope \( \frac{1}{3} \) to find another point (move up 1 unit and 3 units to the right) to get the point (3, 2).Graph \( y = 2 \): - This is a horizontal line passing through y = 2 (y-intercept).

Step 4: Find the intersection point

The intersection point of the two lines is the solution to the system. Identify where the lines intersect on the graph, which is at the point (3, 2).

Key Concepts

Graphing Linear Equations

Graphing linear equations is a way to visually represent relationships between variables. It involves plotting points on a coordinate plane and drawing a line through them.
The general form of a linear equation is often written as: \[ Ax + By = C \]
In graphing, our goal is to make it easier to see where two or more lines intersect. This helps us solve systems of linear equations.

• Each equation represents a line on the coordinate plane.
• Each point on a line is a solution to the equation.

When graphing, it's important to determine key characteristics such as slope and y-intercept to accurately plot the line.

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most common ways to express it. It is written as:
\[ y = mx + b \]
Here, \[ m \] represents the slope of the line, and \[ b \] is the y-intercept.
The slope shows how steep the line is and the direction in which it tilts:

• If the slope is positive, the line rises from left to right.
• If the slope is negative, the line falls from left to right.

The y-intercept is the point where the line crosses the y-axis. When converting an equation to slope-intercept form, you isolate y on one side of the equation so that it fits this format.

Intersection of Lines

The intersection of lines on a graph is significant because it represents the solution to a system of equations. When two lines intersect, it means they share a common point.
To find the intersection point:

• Graph each equation accurately.
• Look for the point where the two lines cross.

This intersection point gives the values of x and y that satisfy both equations simultaneously. In our example, the lines intersect at (3, 2), which is the solution to the system.

Y-Intercept

The y-intercept is a key concept in graphing linear equations. It's the point where the line crosses the y-axis.
For an equation in slope-intercept form \[ y = mx + b \], the y-intercept is represented by \[ b \].
Finding the y-intercept involves setting x to 0 and solving for y:

• In \[ y = \frac{1}{3}x + 1 \], the y-intercept is 1.
• For \[ y = 2 \], it's already y-intercept form, and the y-intercept is 2.

The y-intercept is always indicated as the point (0, b) on the graph. This helps in plotting the line correctly.