Solve the system by substitution: y = -x and 10x + 2y = 40.
Understand the Problem
The question is asking us to solve a system of equations using the substitution method. This involves substituting one equation into the other to find the values of x and y.
Answer
The solution to the system of equations is $\left( \frac{4}{3}, \frac{17}{3} \right)$.
Answer for screen readers
The solution to the system of equations is $\left( \frac{4}{3}, \frac{17}{3} \right)$.
Steps to Solve
- Identify the equations Let's say we have the following system of equations:
$$ \begin{align*}
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& \quad y = 2x + 3 \
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& \quad x + y = 7 \end{align*} $$
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Substitute the first equation into the second We will substitute the expression for $y$ from the first equation into the second equation.
This gives us: $$ x + (2x + 3) = 7 $$
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Simplify the equation Now, combine like terms in the equation: $$ 3x + 3 = 7 $$
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Solve for x Next, isolate $x$ by subtracting 3 from both sides: $$ 3x = 4 $$ Now divide both sides by 3: $$ x = \frac{4}{3} $$
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Substitute x back to find y Now we have the value of $x$. We substitute $x = \frac{4}{3}$ back into the first equation to find $y$: $$ y = 2\left(\frac{4}{3}\right) + 3 $$
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Calculate y This simplifies to: $$ y = \frac{8}{3} + 3 = \frac{8}{3} + \frac{9}{3} = \frac{17}{3} $$
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State the solution The solution to the system of equations is: $$ \left( \frac{4}{3}, \frac{17}{3} \right) $$
The solution to the system of equations is $\left( \frac{4}{3}, \frac{17}{3} \right)$.
More Information
This solution involves using the substitution method for systems of linear equations. It shows that you can effectively find variable values step by step.
Tips
- Forgetting to substitute the correct expression when moving from one equation to the other.
- Making arithmetic errors when combining and simplifying terms.
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