- Rewrite \(x - 2y = 8\) as \(y = \frac{1}{2}x - 4\).
- Rewrite \(-2x + 4y = -16\) as \(y = \frac{1}{2}x - 4\).
- The equations describe the same line, resulting in infinitely many solutions.
- The graph shows coincident lines.
- Slope-intercept form 1: \(y = \frac{1}{2}x - 4\)
- Slope-intercept form 2: \(y = \frac{1}{2}x - 4\)
- Number of solutions: \(\boxed{\text{Infinitely many}}\)
- Graph appearance: \(\boxed{\text{Coincident lines}}\)
Explanation
- Problem Analysis
We are given a system of two linear equations in two variables:
(1) \(x - 2y = 8\)
(2) \(-2x + 4y = -16\)
To analyze these, we will convert both into slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This will allow us to easily compare the lines and determine the number of solutions and their graphical representation.
- Slope-Intercept Form of the First Equation
First, we rewrite equation (1), \(x - 2y = 8\), in slope-intercept form:
Subtract \(x\) from both sides: \(-2y = -x + 8\).
Divide each term by \(-2\):
\(y = \frac{-x}{-2} + \frac{8}{-2}\)
\(y = \frac{1}{2}x - 4\)
Thus, the slope-intercept form of the first equation is \(y = \frac{1}{2}x - 4\).
- Slope-Intercept Form of the Second Equation
Next, we rewrite equation (2), \(-2x + 4y = -16\), in slope-intercept form:
Add \(2x\) to both sides: \(4y = 2x - 16\).
Divide each term by \(4\):
\(y = \frac{2}{4}x - \frac{16}{4}\)
\(y = \frac{1}{2}x - 4\)
Thus, the slope-intercept form of the second equation is also \(y = \frac{1}{2}x - 4\).
- Solutions and Graphical Representation
Since both equations simplify to the exact same expression, \(y = \frac{1}{2}x - 4\), the two lines are identical (coincident).
Because every point on the line \(y = \frac{1}{2}x - 4\) satisfies both equations, there are infinitely many solutions to the system. Graphically, the two lines lie directly on top of each other.
Examples
In real-world scenarios, systems of equations help us understand dependencies. For instance, imagine two different subscription plans for a service that turn out to have the exact same cost structure over time. By modeling these as equations, you can quickly identify if one plan is always cheaper, if they cross at a specific point, or if they are identical, helping you choose the best value. Solving them using the slope-intercept form allows you to easily compare these cost rates (slopes) and starting fees (y-intercepts) visually.