Problem 21 - Algebra A Combined Function

Question

Solve each system of linear equations by graphing. See Examples 3 through \(6 .\) \(\left\\{\begin{array}{l}x+y=5 \\ x+y=6\end{array}\right.\)

Step-by-Step Solution

Answer

The system has no solution because the lines are parallel and do not intersect.

Step 1: Understand the System of Equations

We have a system of two linear equations: \(x + y = 5\) and \(x + y = 6\). Both are linear equations in two variables.

Step 2: Convert Equations into Slope-Intercept Form

Each equation can be written in the form \(y = mx + b\). For \(x + y = 5\), we subtract \(x\) from both sides to get \(y = -x + 5\). For \(x + y = 6\), we also subtract \(x\) from both sides, resulting in \(y = -x + 6\).

Step 3: Graph the Equations

Graph both equations on the same coordinate plane. The first line \(y = -x + 5\) has a y-intercept at \(5\) and a slope of \(-1\), so you go down 1 unit and right 1 unit from the y-intercept. The second line \(y = -x + 6\) has a y-intercept at \(6\) and the same slope of \(-1\), so you again go down 1 unit and right 1 unit from the y-intercept.

Step 4: Interpret the Intersection

Observe where the two lines intersect. In this case, since both lines are parallel (same slope but different y-intercepts), they do not intersect anywhere on the plane.

Key Concepts

Graphing Linear Equations

Graphing linear equations involves plotting a line on a coordinate plane. This line represents all solutions to the equation. When graphing the equation, you need to identify two key components: the slope and the y-intercept.
To graph the equation effectively:

• Begin by identifying the y-intercept, the point where the line crosses the y-axis. For example, in equations like \(y = -x + 5\), the y-intercept is \(5\).
• Then, use the slope (in this case, \(-1\)) to determine direction and steepness. The slope tells us how to move from the y-intercept to another point on the line.

With these two components, plot the y-intercept, apply the slope to find a second point, and draw the line through these points. Make sure the line extends across the coordinate grid.

Parallel Lines

Parallel lines are lines in a plane that never meet. They have the same slope but different y-intercepts. Understanding parallel lines is crucial in solving systems of equations.
A system of equations can result in parallel lines when the equations have:

• Identical slopes (ensuring the lines move in the same direction).
• Different y-intercepts (ensuring the lines are apart).

For example, the lines from the equations \(y = -x + 5\) and \(y = -x + 6\) are parallel because they both have a slope of \(-1\), yet different y-intercepts of \(5\) and \(6\) respectively. This confirms they will never intersect, indicating no common solution when graphed.

Slope-Intercept Form

The slope-intercept form is a way of writing linear equations as \(y = mx + b\). It clearly displays the slope \(m\) and the y-intercept \(b\) of the line.
Key features of the slope-intercept form:

• The slope \(m\) represents the line's direction and steepness. A positive slope rises as it moves from left to right; a negative slope falls.
• The y-intercept \(b\) is where the line crosses the y-axis. This is your starting point on the graph.

For example, when the equation \(x + y = 5\) is rewritten in slope-intercept form as \(y = -x + 5\), it is easy to see the slope \(-1\) and the y-intercept \(5\), thus simplifying the process of graphing the line.

Intersection of Lines

The intersection of lines on a graph represents the solution to a system of linear equations. It's the point where the lines cross, which corresponds to values that satisfy both equations simultaneously.
When graphing two lines, several outcomes are possible:

• If the lines intersect at a single point, that point is the solution to the system.
• If the lines are parallel, as in our example \(x + y = 5\) and \(x + y = 6\), no intersection occurs, indicating no solution to the system.
• If the lines coincide (completely overlap), the system has infinitely many solutions, as any point on the line satisfies both equations.

In the provided system of equations, since the lines are parallel, no intersection point exists, thus illustrating a system with no solutions.