Question
6-85. Jai was solving the system of equations below when something strange happened. \[ y=-2 x+5 \] \( 2 y+4 x=10 \) a. Solve the system. What happened? b. Graph the two lines on the same set of axes. What happened? \( \square \square \) Hint (b): c. How does your graph in part (b) explain your result in part (a)? \( \square \) Hint (c):
Ask by Daniel Stephens. in the United States
Mar 01,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
Part (a): The system has infinitely many solutions because both equations represent the same line.
Part (b): When graphed, both lines overlap completely, showing they are the same line.
Part (c): The overlapping lines mean both equations are equivalent, resulting in infinitely many solutions.
Solution
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
\(\left\{ \begin{array}{l}y=-2x+5\\2y+4x=10\end{array}\right.\)
- step1: Substitute the value of \(y:\)
\(2\left(-2x+5\right)+4x=10\)
- step2: Simplify:
\(10=10\)
- step3: Calculate:
\(\left(x,y\right) = \left(x,-2x+5\right),x \in \mathbb{R}\)
- step4: Alternative Form:
\(\textrm{Infinitely many solutions}\)
Let's solve the problem step by step.
### Part (a): Solve the system of equations
We have the following system of equations:
1. \( y = -2x + 5 \)
2. \( 2y + 4x = 10 \)
To solve this system, we can substitute the expression for \( y \) from the first equation into the second equation.
Substituting \( y \) in the second equation:
\[
2(-2x + 5) + 4x = 10
\]
Now, let's simplify this equation:
\[
-4x + 10 + 4x = 10
\]
This simplifies to:
\[
10 = 10
\]
This is a true statement, which means that the two equations represent the same line. Therefore, there are infinitely many solutions to this system of equations.
### Part (b): Graph the two lines
Since I cannot plot the graph directly, I can describe what happens when you graph these two equations:
1. The first equation \( y = -2x + 5 \) is a straight line with a slope of -2 and a y-intercept of 5.
2. The second equation \( 2y + 4x = 10 \) can be rewritten as \( y = -2x + 5 \) after simplification, which is the same line as the first equation.
When you graph both lines, they will overlap completely, indicating that they are the same line.
### Part (c): Explanation of the graph in relation to part (a)
The graph shows that both equations represent the same line, which explains the result in part (a). Since the two equations are equivalent, they have infinitely many solutions, as every point on the line is a solution to both equations.
In summary:
- **Part (a)**: The system has infinitely many solutions.
- **Part (b)**: The graph of both equations is the same line.
- **Part (c)**: The overlapping lines confirm that the equations are equivalent, leading to infinitely many solutions.
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Simplify this solution Beyond the Answer
The two equations represent lines in a two-dimensional space. When we simplify the second equation, \(2y + 4x = 10\), by dividing everything by 2, we get \(y + 2x = 5\), or rearranging it gives us \(y = -2x + 5\). This shows us that both equations are actually the same line. Therefore, the system has infinitely many solutions, as every point on the line is a solution. When you graph both equations, you'll notice that they lie exactly on top of each other, illustrating that they are the same line. This visually confirms that there is not just one solution but an infinite number since every point on that line satisfies both equations. Thus, the two lines coincide perfectly!
