Describe how you would obtain an equation in one variable to solve the system by substitution.
Understand the Problem
The question is asking for a description of the process to obtain a single-variable equation from the given system of equations in order to use substitution for solving them.
Answer
The single-variable equation obtained is $7y = 7$, leading to $y = 1$.
Answer for screen readers
The single-variable equation after substitution is $7y = 7$, which simplifies to $y = 1$.
Steps to Solve
- Isolate one variable from the first equation
From the first equation $x - y = -3$, we can isolate $x$. Rearranging gives: $$ x = y - 3 $$
- Substitute the isolated variable into the second equation
Now substitute $x$ in the second equation $4x + 3y = -5$. We replace $x$ with $y - 3$: $$ 4(y - 3) + 3y = -5 $$
- Simplify the equation
Distributing the 4: $$ 4y - 12 + 3y = -5 $$
Combine like terms: $$ 7y - 12 = -5 $$
- Isolate the remaining variable
Now, isolate $y$ by adding 12 to both sides: $$ 7y = 7 $$
- Solve for the variable
Finally, divide by 7: $$ y = 1 $$
The single-variable equation after substitution is $7y = 7$, which simplifies to $y = 1$.
More Information
After finding $y = 1$, you can substitute back into $x = y - 3$ to find $x$. Therefore, $x = 1 - 3 = -2$. Thus, the solution to the system is $x = -2$ and $y = 1$.
Tips
- Incorrectly substituting the variable: Always ensure you substitute the right value into the correct equation.
- Neglecting to distribute properly: Pay attention to distributing values correctly across terms.
- Simplifying incorrectly: Check your addition and subtraction when combining like terms.
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