Problem 21 - Introductory Algebra for College Students

Question

Solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l}y=2 x \\ y=-x+6\end{array}\right.$$

Step-by-Step Solution

Answer

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

Step 1: Graph the first equation

Let's start by graphing the first equation \(y=2x\). This is a straight line that passes through the origin, and for every unit increase in \(x\), \(y\) increases by 2 units, since the slope is positive and equal to 2. This line will be upward-sloping.

Step 2: Graph the second equation

Next, we need to graph the second equation, \(y=-x+6\). This line intersects the y-axis at \(y=6\) (the y-intercept) and for each unit increase in \(x\), \(y\) decreases by 1 unit (because the slope is negative and equal to -1). So, this line will be downward-sloping.

Step 3: Find the intersection

After drawing both lines, examine where they intersect. This point represents the solution for this system of linear equations. Here, the intersection is at the point (2,4). It means \(x=2\) and \(y=4\) is the solution to the system.

Step 4: Express the solution in set notation

Finally, as requested in the exercise, we express the answer in set notation. The solution can be written as \{(2, 4)\}.

Key Concepts

Linear Equations

In mathematics, a linear equation is a straightforward, two-variable equation represented in the form of: \[ y = mx + b \] where:

• \(m\) is the slope of the line.
• \(b\) represents the y-intercept.

The slope determines how steep the line is, and the y-intercept indicates where the line crosses the y-axis. A linear equation forms a straight line when graphed on a coordinate grid. Such equations describe a constant rate of change, indicating that their graphs will always be straight lines. Understanding this form helps to graph the line quickly, as the two essential pieces of knowledge—centre the y-intercept on the y-axis and then use the slope to find another point.

Solution Sets

A solution set is a collection of all the solutions that satisfy a given equation or system of equations. For linear equations that intersect in a single point, the solution set is typically represented by the point of intersection. For example, given the system of equations:

• \( y = 2x \)
• \( y = -x + 6 \)

The solution set comprises the coordinates where both these lines intersect. In this particular instance,

• The lines intersect at the point (2, 4),
• making the solution set \[ \{(2, 4)\} \].

For a system of equations, the solution set expresses all the potential solutions to the problems in the form of ordered pairs (\(x, y\)). For unique intersections, the solution is distinct. If there were infinite solutions, the lines would be overlapping, and no solution would mean they are parallel.

Intersection of Lines

The intersection of lines in a graph occurs at the point where two lines meet. In the context of linear equations, this intersection represents the values of \(x\) and \(y\) that satisfy both equations simultaneously. Finding the intersection:

• The intersection represents the graphical solution to a system of equations.
• Intersections are determined by solving the equations in the system for both \(x\) and \(y\).
• Graphically, it's the point at which both lines cross each other.

In the example of the system of equations:

• \( y = 2x \)
• \( y = -x + 6 \)

The intersection occurs at the point (2, 4). This point is significant because it confirms the simultaneous truth of both equations, thus verifying the solution through their graphical representation. Understanding intersection points is crucial for solving and visualizing systems of linear equations efficiently.