- Substitute \(y = 4x + 1\) into \(2y = -x - 1\) to get \(2(4x + 1) = -x - 1\).
- Expand the equation to \(8x + 2 = -x - 1\).
- Simplify by moving all \(x\) terms to one side and constants to the other, yielding \(8x + x = -1 - 2\).
- The final simplified equation is \(9x = -3\), which is option c).
- Final Answer: \(\boxed{c)}\)
Explanation
- Problem Analysis
We are given a system of two linear equations:
- \(y = 4x + 1\)
- \(2y = -x - 1\)
Our goal is to substitute the expression for \(y\) from the first equation into the second equation to produce a single linear equation in terms of \(x\). We will then determine which of the provided options matches the resulting equation.
- Substitution and Expansion
The method of substitution involves replacing a variable in one equation with its equivalent expression from another equation. Here, we know that \(y = 4x + 1\). We substitute this into the second equation, \(2y = -x - 1\), by replacing \(y\) with \(4x + 1\):
\[2(4x + 1) = -x - 1\]
Next, we distribute the \(2\) on the left side of the equation:
\[2 \cdot 4x + 2 \cdot 1 = -x - 1\]
\[8x + 2 = -x - 1\]
- Simplification and Solving for x
Now, we rearrange the equation to group all terms containing \(x\) on one side and all constants on the other side. Add \(x\) to both sides and subtract \(2\) from both sides:
\[8x + x = -1 - 2\]
Combining like terms on both sides gives us:
\[9x = -3\]
- Final Answer Selection
Comparing our resulting equation \(9x = -3\) with the given choices:
A\(9x = 1\)
B\(7x = 1\)
C\(9x = -3\)
D\(7x = -3\)
We can see that our result matches option c). Thus, the correct equation is \(9x = -3\), which corresponds to option c). The final answer is \(\boxed{c)}\).
Examples
Substitution is a fundamental technique in algebra for solving systems of linear equations. This method is frequently applied in real-life scenarios like resource management or cost analysis. For example, if you are comparing two different payment plans for a service—where one plan has a fixed cost and a variable rate and the other is a different combination—you can use substitution to find the exact point at which both plans cost the same amount, helping you make an informed decision on which option is more economical.