Use elimination by substitution to solve the system: x + 2y - 8 = 0; 2x + 4y + 4 = 0.
Understand the Problem
The question is asking to use the method of elimination by substitution to solve a system of equations. It leads us through the steps to find the solution, demonstrating that there is no solution since the resulting equation is never true.
Answer
The system of equations has no solution.
Answer for screen readers
The system of equations has no solution.
Steps to Solve
-
Identify the equations
We have the following system of equations:
$$
\begin{cases}
x + 2y - 8 = 0 \quad (1) \
2x + 4y + 4 = 0 \quad (2)
\end{cases}
$$ -
Solve the first equation for (x)
From equation (1), isolate (x):
$$
x = -2y + 8
$$ -
Substitute into the second equation
Now, substitute (x = -2y + 8) into equation (2):
$$
2(-2y + 8) + 4y + 4 = 0
$$ -
Expand and simplify
Distributing the (2):
$$
-4y + 16 + 4y + 4 = 0
$$
Combine like terms:
$$
20 = 0
$$
- Analyze the result
The equation (20 = 0) is never true, indicating that the system has no solution.
The system of equations has no solution.
More Information
This situation occurs when the two lines represented by the equations are parallel, meaning they never intersect. In this case, both lines have the same slope but different y-intercepts.
Tips
- Miscalculating the substitution can lead to incorrect equations.
- Neglecting to check if the derived equation is true for any values of (y) can lead to false conclusions.
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