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Magazine Chapter 4: Problem 43
Solve each system of equations by graphing. \(x-2 y=10\) \(2 x-4 y=12\)
Short Answer
Expert verified
The system has no solution; the lines are parallel.
Step by step solution
01
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 \(y = \frac{1}{2}x - 3\).
02
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.
03
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.
04
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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.
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.
No Solution
In systems of equations, a solution exists where the lines intersect鈥攚hich 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.
