Problem 43 - College Algebra with Corequisite Support

Question

For the following exercises, graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions. $$ \begin{array}{r} x+2 y=7 \\ 2 x+6 y=12 \end{array} $$

Step-by-Step Solution

Answer

The system is consistent and has one solution.

Step 1: Write Equations in Slope-Intercept Form

Convert each equation into the form \( y = mx + c \). For the first equation \( x + 2y = 7 \), subtract \( x \) from both sides to get \( 2y = -x + 7 \). Then divide everything by 2 to obtain \( y = -\frac{1}{2}x + \frac{7}{2} \). For the second equation \( 2x + 6y = 12 \), subtract \( 2x \) from both sides yielding \( 6y = -2x + 12 \). Divide by 6 to get \( y = -\frac{1}{3}x + 2 \).

Step 2: Graph Each Equation

Draw the graph for each equation on a coordinate plane. The first equation \( y = -\frac{1}{2}x + \frac{7}{2} \) is a line with slope \(-\frac{1}{2}\) and y-intercept at \( \left(0, \frac{7}{2}\right) \). The second equation \( y = -\frac{1}{3}x + 2 \) is a line with slope \(-\frac{1}{3}\) and y-intercept at \( (0, 2) \).

Step 3: Analyze the Graphs

Look at where the lines intersect. Since the lines have different slopes, they intersect at exactly one point, which means the system of equations is consistent. A consistent system with distinct slopes has exactly one solution.

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the most popular ways to express linear equations. It makes it straightforward to identify the key components of a line: the slope and the y-intercept. The slope-intercept form is given by the equation \( y = mx + c \). Here, \( m \) represents the slope and \( c \) represents the y-intercept. This format is particularly beneficial because it allows us to easily determine how one variable changes in relation to another.

To convert equations into this form, you basically make \( y \) the subject of the formula. For example,

• If you have an equation like \( x + 2y = 7 \), you would first isolate \( y \) by moving \( x \) to the other side, resulting in \( 2y = -x + 7 \).
• Then, you divide all parts by 2, simplifying to \( y = -\frac{1}{2}x + \frac{7}{2} \).

This process is the same for all linear equations and is an essential skill when dealing with graphing equations and solving systems. It's the first step to understanding the behavior of the line associated with an equation.

Consistent Systems

In the world of systems of equations, all systems fall into one of three categories: consistent, inconsistent, or dependent. Understanding these is crucial for solving systems of equations effectively.

A consistent system is when the equations share at least one common solution. This means they either intersect at a single point in the case of two different slopes, or are the same line, resulting in infinite solutions. In consistent systems, you'll find one or more solutions that satisfy both equations.

• One Solution: This occurs when the lines have different slopes and intersect at one distinct point.
• Infinite Solutions: This happens when the lines are essentially the same, overlapping completely.

In the example given, where we have equations like \( y = -\frac{1}{2}x + \frac{7}{2} \) and \( y = -\frac{1}{3}x + 2 \), the slopes are different (\(-\frac{1}{2} \) and \(-\frac{1}{3} \)), indicating that the system is consistent with exactly one solution. This means the lines intersect at only one point on a graph.

Graphing Linear Equations

Graphing linear equations is an essential skill in algebra, allowing you to visually interpret the equations and understand their solutions better. A graph provides a picture of the relationships between variables.

To graph an equation, you can use the slope-intercept form (\( y = mx + c \)), as it readily shows you the slope and y-intercept. Here’s how you can do it:

• Identify the y-intercept (\( c \)). This tells you where the line crosses the y-axis, providing your starting point.
• Use the slope (\( m \)), which represents rise over run, to find another point starting from the y-intercept.

For instance:

• With the equation \( y = -\frac{1}{2}x + \frac{7}{2} \), start at the point \((0, \frac{7}{2})\) on the y-axis.
• From this point, use the slope \(-\frac{1}{2}\) to find another point by going down 1 unit and right 2 units.

Once you've plotted these points, draw a straight line through them. Repeat the steps for the other equation.

Graphing allows you to clearly see if the lines intersect and where they intersect, revealing the solution to the system of equations when they have different slopes.