Problem 43 - Algebra 2

Question

Solve each system of equations by graphing. \(x-2 y=10\) \(2 x-4 y=12\)

Step-by-Step Solution

Answer

The system has no solution; the lines are parallel.

Step 1: Convert Equations to Slope-Intercept Form

First, we need to write both equations in the slope-intercept form \(y = mx + b\). This form is helpful for graphing. Starting with the first equation \(x - 2y = 10\), solve for \(y\): \(-2y = -x + 10\) which simplifies to \(y = \frac{1}{2}x - 5\). For the second equation \(2x - 4y = 12\), solve for \(y\): \(-4y = -2x + 12\), then simplify to \(y = \frac{1}{2}x - 3\).

Step 2: Graph the First Equation

Graph the equation \(y = \frac{1}{2}x - 5\). The y-intercept is \(-5\), so plot the point \((0, -5)\). The slope \(\frac{1}{2}\) indicates a rise of 1 unit for every run of 2 units. From \((0, -5)\), move up 1 unit and right 2 units to plot another point, then draw the line through these points.

Step 3: Graph the Second Equation

Graph the equation \(y = \frac{1}{2}x - 3\). The y-intercept is \(-3\), so plot the point \((0, -3)\). The slope \(\frac{1}{2}\) again indicates a rise of 1 unit for every 2 units run. From \((0, -3)\), move up 1 unit and right 2 units to plot another point, then draw the line through these points.

Step 4: Analyze the Graphs

Observe the two lines. They are parallel since they have the same slope (\(\frac{1}{2}\)) but different y-intercepts (\(-5\) and \(-3\)). Since they never intersect, the system of equations has no solution.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept. This form is convenient for graphing as it provides straightforward values:

• Slope \( (m) \): Shows the steepness of the line and its direction. A positive slope means the line goes uphill, while a negative slope goes downhill.
• Y-intercept \( (b) \): Indicates where the line crosses the y-axis. This is your starting point when plotting a line on a graph.

Using the slope-intercept form, you can easily convert complex equations to this simple representation, facilitating the graphing process. For example, converting \( x - 2y = 10 \) to slope-intercept form helps us see its characteristics immediately.

Graphing

Graphing is a visual way to represent mathematical equations, especially helpful for solving systems of equations. First, identify the y-intercept and slope from the slope-intercept form. Ensure to:

• Start by plotting the y-intercept, \( (0, b) \), on the graph. This gives you a concrete point to begin.
• Utilize the slope \( m = \frac{rise}{run} \) to find another point. From the y-intercept, use the slope to rise (up or down) and run (right or left) to place your next point.

Connect these points to draw your line. The intersection of lines in a graph indicates the solution of the system of equations, though in some cases like parallel lines, there may be no intersection.

Parallel Lines

Parallel lines are lines in a plane that never intersect. In the context of linear equations:

• They have the same slope \( m \), ensuring they move in the same direction.
• They have different y-intercepts \( b \), which positions them at different vertical places on the graph.

For our exercise, both equations \( y = \frac{1}{2}x - 5 \) and \( y = \frac{1}{2}x - 3 \) have the same slope \( \frac{1}{2} \), confirming they are parallel. The different y-intercepts (-5 and -3) mean they are offset from each other vertically but never cross.

No Solution

In systems of equations, a solution exists where the lines intersect—which represents a shared solution for both equations. However, with parallel lines:

• They never intersect, thus there is no point that satisfies both equations.
• This situation results in a "no solution" outcome for the system.

This is the case with our exercise, where the parallel lines do not meet, indicating that no solution exists for the given set of equations. Understanding when systems have no solution is crucial in solving linear equations, helping avoid misinterpretations.