Problem 19 - Elementary Algebra Within Reach

Question

In Exercises 19-26, solve the system by graphing. $$ \left\\{\begin{array}{r} x+y=4 \\ 4 x+4 y=2 \end{array}\right. $$

Step-by-Step Solution

Answer

The system of equations has no solution as both the lines represented by the equations are parallel.

Step 1: Write the Equations in Slope-Intercept Form

The slope-intercept form of an equation is given by \(y = mx+c\), where m is the slope and c is the y-intercept. The given system can be rewritten in slope-intercept form as \(y = -x + 4\) and \(y = -x + 0.5\).

Step 2: Graph the Equations

Graph the two equations on the same set of axes. The straight line from the equation \(y = -x + 4\) intersects the y-axis at 4 and has a slope of -1. Similarly, the line from the equation \(y = -x + 0.5\) intersects the y-axis at 0.5 and also has a slope of -1. Draw these two lines on the graph paper.

Step 3: Find the Intersection Point

After plotting both lines, it can be seen that these lines are parallel and hence, they do not intersect. A pair of parallel lines has no common point, indicating that the system of equations has no solution.

Key Concepts

Slope-Intercept Form

When solving systems of equations by graphing, the slope-intercept form is a useful tool. The equation takes the shape of \( y = mx + c \), where \( m \) is the slope, and \( c \) is the y-intercept.
This form is convenient because it gives a clear picture of how the line behaves on a graph. By simply knowing the values of \( m \) and \( c \), you can easily draw the line.

• **Slope (m):** The slope describes the line's steepness and direction. A positive slope means the line rises as you move from left to right, while a negative slope indicates it falls. The slope of our lines in the exercise is -1, meaning they are decreasing.
• **Y-intercept (c):** This is the point where the line crosses the y-axis. For instance, in \( y = -x + 4 \), the line crosses the y-axis at 4.

Using the slope-intercept form helps you quickly plot each equation on the graph, making the visual comparison straightforward.

Parallel Lines

Parallel lines are lines in a plane that never meet. They have the same slope but different y-intercepts, which means they run at an equal distance to each other. In our exercise, both equations have the slope of -1, showing that the lines they represent are indeed parallel.
Parallel lines are an indicator of specific behaviors in systems of equations. Here are some important aspects:

• If two lines are parallel, their equations will have the same slope \( m \) but different y-intercepts \( c \).
• On a graph, parallel lines will never cross each other, and thus they'll never share any points.

This quality of parallel lines is crucial in determining the nature of the solutions of system of equations. In this case, since our lines are parallel, they do not have a shared solution.

No Solution in System of Equations

When dealing with a system of equations that graph as parallel lines, the system will have no solutions. This is because parallel lines never intersect, meaning there is no point that satisfies both equations simultaneously.
A system with no solution is known as "inconsistent". Here are key features of such systems:

• **Parallelism:** The system will generate lines that have identical slopes but distinct y-intercepts.
• **Graph Interpretation:** When graphed, it is easy to see that there are no intersections, reinforcing that no solutions exist.
• **Forming Equations:** Often, algebraic manipulation (like getting the slope-intercept form) can expose parallel nature before graphing.

Graphing is a visual way to identify when no solutions exist. If your equations in slope-intercept form result in equal slopes, be ready to conclude parallelism and no shared solutions.