Problem 20 - Algebra 1: Concepts and Skills

Question

Use the graphing method to tell how many solutions the system has. $$\begin{aligned} -x+4 y &=-20 \\ 3 x-12 y &=48 \end{aligned}$$

Step-by-Step Solution

Answer

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

Step 1: Reformulate Equations

Reformulate both equations into the slope-intercept form \(y = mx + b\): \[ \begin{aligned}\ -x+4y = -20 & \ y = \frac{1}{4}x + 5 \end{aligned}\] and \[ \begin{aligned}\ 3x-12y = 48 & \ y = \frac{1}{4}x - 4 \end{aligned}\]

Step 2: Graph the Equations

Now, graph both lines on the same graph. The line \(y = \frac{1}{4}x + 5\) has a slope of \frac{1}{4} and a y-intercept of 5. The line \(y = \frac{1}{4}x - 4\) also has a slope of \frac{1}{4}, but a y-intercept of -4.

Step 3: Analyze the Intersection Points

By inspecting the graphs, it can be observed that the lines do not intersect at all. This is because they have the same slopes but different y-intercepts; therefore, they are parallel to each other.

Key Concepts

Graphing Method

The graphing method is a visual way to solve systems of equations by plotting each equation on a coordinate plane and identifying the point where they intersect. This method can be very insightful, especially for understanding the relationships between equations. It provides a clear graphical representation of solutions.

• To use this technique, first convert each equation into a form that's easy to graph, typically the slope-intercept form.
• Plot the lines corresponding to each equation, using key points such as the y-intercept and slope to draw them accurately.
• Observe where the lines meet. The coordinates of the intersection point are the solutions to the system.

If the lines intersect at one point, the system has a single unique solution. If they coincide (are the same line), there are infinitely many solutions. If the lines are parallel, they never intersect, meaning there are no solutions. Graphing gives a straightforward way of determining this visually.

Slope-Intercept Form

Converting equations to the slope-intercept form is often the first step in the graphing method. This form of a linear equation is expressed as \(y = mx + b\), where:

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

Converting to this form simplifies plotting the line on a graph:- Simply locate the y-intercept (where the line crosses the y-axis) on the graph, which is \(b\).
- Use the slope \(m\) to determine the next set of points. The slope is a ratio: \(\text{rise}/\text{run}\), indicating how many units you go up or down for each unit you move to the right.
The consistency in representation through the slope and y-intercept allows for easy comparison of multiple lines, helping identify parallel lines or intersection points quickly.

Parallel Lines

Parallel lines in a coordinate plane have the same slope but different y-intercepts. This characteristic implies that they will never meet, no matter how far they are extended. Identifying parallel lines is crucial in determining solutions for a system of equations:

• If two lines have identical slopes (\(m_1 = m_2\)) but different y-intercepts, they are parallel.
• Graphically, parallel lines never intersect, which means there is no common solution to the system of equations they represent.
• For example, in the exercise, both lines have the slope \(\frac{1}{4}\) but different y-intercepts (5 and -4), confirming they're parallel.

Understanding the nature of parallel lines helps in visualizing situations where issues like no solutions arise in systems of equations. It also simplifies predicting outcomes just by inspecting the algebraic expressions.