Problem 15 - Elementary Algebra

Question

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing. $$ \left\\{\begin{array}{l} y=\frac{3}{2} x+1 \\ y=-\frac{1}{2} x+5 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system is (2, 4).

Step 1 - Understand the equations

Identify the equations given in the system: \( y = \frac{3}{2} x + 1 \) and \( y = -\frac{1}{2} x + 5 \).

Step 2 - Plot the first equation

To graph \( y = \frac{3}{2} x + 1 \), start by plotting the y-intercept at (0, 1). Then use the slope to find another point: from (0,1), move up 3 units and 2 units to the right to reach the point (2, 4). Draw a line through these points.

Step 3 - Plot the second equation

To graph \( y = -\frac{1}{2} x + 5 \), start by plotting the y-intercept at (0, 5). Then use the slope to find another point: from (0,5), move down 1 unit and 2 units to the right to reach the point (2, 4). Draw a line through these points.

Step 4 - Identify the intersection point

Observe the graph. The point where the two lines intersect is the solution to the system of equations. The lines intersect at the point (2, 4).

Key Concepts

Graphing Linear Equations

To solve a system of linear equations by graphing, you first need to understand what a linear equation is. A linear equation is one where the graph forms a straight line. The general form is: \( y = mx + b \). Here, \( m \) represents the slope, and \( b \) represents the y-intercept.

Let's break this down using the given equations: First Equation: \( y = \frac{3}{2} x + 1 \)
Second Equation: \( y = -\frac{1}{2} x + 5 \)

For the first equation, the slope \( m \) is \( \frac{3}{2} \) and the y-intercept \( b \) is 1. For the second, the slope is \( -\frac{1}{2} \) and y-intercept is 5.

Next, plot these equations on a graph. Start with the y-intercept and use the slope to find the next point. The slope \( \frac{3}{2} \) tells you to move up 3 units and 2 units to the right. This gives you the point (2, 4). For the second equation, starting from (0, 5), move down 1 unit and 2 units to the right, which also gives you (2, 4).

Solving Equations

When we talk about solving a system of linear equations, we aim to find the values of variables that satisfy all equations simultaneously. This is often represented as finding the point where two lines intersect on a graph.

Let's go through this step-by-step for the given system: Step 1: Identify the equations: \( y = \frac{3}{2} x + 1 \) and \( y = -\frac{1}{2} x + 5 \).
Step 2: Plot the first equation using its slope and y-intercept.
Step 3: Plot the second equation similarly.
Step 4: Observe where the lines intersect.

The intersection point is the solution to the system of equations. If you correctly graph both lines, they will meet at a single point, which represents the solution. If the lines do not intersect, then the system has no solution. If the lines overlap completely, then there are infinitely many solutions.

Intersection Point

The intersection point is crucial in solving a system of linear equations by graphing. It is the point where the two lines cross each other and represents the values of \( x \) and \( y \) that satisfy both equations at the same time.

Using our current example, after graphing \( y = \frac{3}{2} x + 1 \) and \( y = -\frac{1}{2} x + 5 \), we found that the lines intersect at (2, 4). Here is why this is important:

• At point (2, 4), both equations hold true because substituting \( x = 2 \) into both equations results in \( y = 4 \).
• This means we have found the solution to the system of linear equations.
• Without the intersection, there would be no common solution, and hence no point that satisfies both equations simultaneously.

Understanding the concept of intersection points can make solving systems of linear equations more intuitive and visually clear.