Find the solution of the system of equations: 3x + 3y = 15 and 12x + 8y = 20.
Understand the Problem
The question is asking for the solution to a system of equations consisting of two linear equations. We will need to apply methods like substitution or elimination to find the values of the variables that satisfy both equations.
Answer
The solution is \((x, y) = (-5, 10)\).
Answer for screen readers
The solution to the system of equations is ((x, y) = (-5, 10)).
Steps to Solve
- Identify the equations
The system of equations is: $$ 3x + 3y = 15 $$ $$ 12x + 8y = 20 $$
- Simplify the first equation
Divide the entire first equation by 3 to simplify it: $$ x + y = 5 $$
- Express one variable in terms of the other
Rearrange the simplified equation to find $y$ in terms of $x$: $$ y = 5 - x $$
- Substitute into the second equation
Substitute $y$ in the second equation: $$ 12x + 8(5 - x) = 20 $$
- Solve for $x$
Distribute the $8$ in the equation: $$ 12x + 40 - 8x = 20 $$
Combine like terms: $$ 4x + 40 = 20 $$
Subtract $40$ from both sides: $$ 4x = -20 $$
Divide by 4: $$ x = -5 $$
- Find $y$ using the value of $x$
Substitute $x = -5$ back into the expression for $y$: $$ y = 5 - (-5) $$ $$ y = 10 $$
- Write the final solution
The solution to the system is: $$(x, y) = (-5, 10)$$
The solution to the system of equations is ((x, y) = (-5, 10)).
More Information
This solution can be verified by substituting (x) and (y) back into the original equations, confirming that both equations hold true.
Tips
- Failing to simplify equations which can lead to more complex calculations.
- Misapplying distribution during substitution, which can change the terms.
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