Use elimination by substitution to solve the system: x + 2y - 8 = 0 and 2x + 4y + 4 = 0.
Understand the Problem
The question is asking to use the elimination method by substitution to solve a system of equations. The provided example explains the process of isolating one variable and substituting it into the other equation to find a solution or determine if no solution exists.
Answer
There is no solution to the system of equations.
Answer for screen readers
There is no solution to the given system of equations.
Steps to Solve
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Isolate one variable from the first equation
Start with the first equation: $$ x + 2y - 8 = 0 $$
Rearranging gives: $$ x = -2y + 8 $$
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Substitute in the second equation
Take the expression for $x$ and substitute it in the second equation: $$ 2x + 4y + 4 = 0 $$
Replacing $x$ yields: $$ 2(-2y + 8) + 4y + 4 = 0 $$
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Simplify the equation
Distributing gives: $$ -4y + 16 + 4y + 4 = 0 $$
Combining like terms results in: $$ 20 = 0 $$
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Analyze the result
The equation $20 = 0$ is never true, indicating that there is no solution to the system of equations.
There is no solution to the given system of equations.
More Information
This system represents two lines that are parallel, as they have the same slope of $-\frac{1}{2}$ but different y-intercepts (4 and -1). Parallel lines never intersect, indicating there is no solution.
Tips
- Failing to properly isolate the variable from the first equation, which can lead to incorrect substitution.
- Not checking the simplified equation for valid solutions, particularly recognizing inconsistent statements like $20 = 0$.
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