Problem 51 - College Algebra and Calculus: An Applied Approach

Question

Use a graphing utility to determine whether the system of equations has one solution, two solutions, or no solution. $$\left\\{\begin{array}{l}y=-5 x+1 \\ y=x+3\end{array}\right.$$

Step-by-Step Solution

Answer

When the lines of the equations are graphed, they intersect at one point, which means that the system of equations has one solution.

Step 1: Plot the first equation

First, plot the line of the equation \( y = -5x + 1 \). This is a linear equation, so it will be a straight line when graphed.

Step 2: Plot the second equation

Next, plot the line of the equation \( y = x + 3 \). This is also a linear equation, and will be another straight line when graphed.

Step 3: Identify the Intersection

Look at where the lines from Step 1 and Step 2 intersect. If they intersect at a single point, this is the solution to the system. If the lines do not intersect, then the system has no solutions. If the lines are the same line (they coincide), then there are infinite solutions.

Key Concepts

Linear Equations

Linear equations form the backbone of algebra. A linear equation is a type of equation where the highest power of the variable is one. In the context of our exercise, each equation represents a straight line when plotted on a graph. The general form for a linear equation in two variables is \( y = mx + b \), where:

• \( m \) is the slope of the line, indicating its steepness or tilt.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

Both equations in the exercise, \( y = -5x + 1 \) and \( y = x + 3 \), fit this form. The differences in their slopes and intercepts determine how these lines behave relative to each other when graphed.

Graphing Utility

A graphing utility is a tool, often digital, that assists in plotting equations on a coordinate plane. These tools are incredibly helpful when dealing with systems of equations, as they provide a visual representation of the relationships between equations.

When you input the equations \( y = -5x + 1 \) and \( y = x + 3 \) into a graphing utility, it will graph each line based on their respective slopes and y-intercepts. The visualization helps in easily identifying where the lines meet, if at all.

• Plot the equations accurately to see where they may intersect.
• Different colors or styles may be used to distinguish between the lines.

Using a graphing utility can save time and reduce errors in plotting, especially with more complex equations.

Intersection

The intersection of two lines refers to the point where they cross each other on a graph. In terms of solving a system of linear equations, finding the intersection is crucial as it represents the solution to the system.

For our given equations, \( y = -5x + 1 \) and \( y = x + 3 \), the intersection point can be found by plotting both lines. The coordinates of this meeting point are the values of \( x \) and \( y \) that satisfy both equations simultaneously.

• One intersection means there is a unique solution to the system.
• No intersection implies the system has no solutions.
• Countless intersections mean the lines are identical, indicating infinitely many solutions.

Solution of a System

The solution of a system of equations is the set of values that satisfy all equations involved. For linear equations, this solution is understood graphically as the intersection point(s) of the lines.

In our example with the system of equations \( y = -5x + 1 \) and \( y = x + 3 \), the solution is the point where these two lines intersect. By visualizing the system on a graphing utility, we identify the solution by seeing a single point where the lines cross.

• If lines intersect at one point, the system has a singular solution.
• If the lines run parallel without crossing, there’s no solution.
• If lines overlap entirely, the solution includes all points on the line, indicating infinite solutions.

Understanding the solution helps in solving real-world problems where multiple conditions need to be met simultaneously.