Solve the system by substitution: 3x + 10y = 31, 2x + 10 = y
Understand the Problem
The question is asking us to solve a system of equations using the substitution method. We need to find the values of x and y that satisfy both equations provided.
Answer
$(-3, 4)$
Answer for screen readers
The solution to the system of equations is $(-3, 4)$.
Steps to Solve
- Solve for y in the second equation
From the second equation, we can express $y$ in terms of $x$:
$$ y = 2x + 10 $$
- Substitute y in the first equation
Now replace $y$ in the first equation with the expression we found:
$$ 3x + 10(2x + 10) = 31 $$
- Simplify the equation
Distribute $10$ in the equation:
$$ 3x + 20x + 100 = 31 $$
Combine like terms:
$$ 23x + 100 = 31 $$
- Isolate x
Subtract $100$ from both sides:
$$ 23x = 31 - 100 $$
Which simplifies to:
$$ 23x = -69 $$
Now divide by $23$:
$$ x = -3 $$
- Find y using x
Substitute $x = -3$ back into the equation we found for $y$:
$$ y = 2(-3) + 10 $$
This results in:
$$ y = -6 + 10 $$
Finally, we have:
$$ y = 4 $$
The solution to the system of equations is $(-3, 4)$.
More Information
The solution $(-3, 4)$ means that when $x = -3$, $y = 4$ are the values that satisfy both original equations simultaneously. This method demonstrates the substitution method, which is useful for solving linear systems efficiently.
Tips
- Incorrect substitution: Ensure substitution is done accurately to avoid errors in the equations.
- Combining terms incorrectly: Pay attention to combining like terms carefully, as mistakes can lead to incorrect results.
AI-generated content may contain errors. Please verify critical information
