Solve the system of linear equations by substitution. 8x = 5y + 24, -9y = 40 - 8x. The solution is: ( , )
Understand the Problem
The question is asking to solve a system of linear equations using substitution. This involves isolating one variable in one equation and substituting it into the other equation to find the values of both variables.
Answer
The solution is \( \left( \frac{1}{2}, -4 \right) \)
Answer for screen readers
The solution is: ( \left( \frac{1}{2}, -4 \right) )
Steps to Solve
- Isolate one variable in the first equation
Start with the first equation: $$ 8x = 5y + 24 $$
Rearranging it to isolate $y$: $$ 5y = 8x - 24 $$ $$ y = \frac{8x - 24}{5} $$
- Substitute the expression into the second equation
Now, substitute the expression for $y$ into the second equation: $$ -9y = 40 - 8x $$
Substituting $y$: $$ -9\left(\frac{8x - 24}{5}\right) = 40 - 8x $$
- Clear the fraction and simplify
Multiply both sides by 5 to eliminate the fraction: $$ -9(8x - 24) = 5(40 - 8x) $$
Distributing gives: $$ -72x + 216 = 200 - 40x $$
- Move all terms involving x to one side
Add $72x$ to both sides: $$ 216 = 200 + 32x $$
Subtract 200 from both sides: $$ 16 = 32x $$
- Solve for x
Divide both sides by 32: $$ x = \frac{16}{32} = \frac{1}{2} $$
- Substitute x back to find y
Substitute $x = \frac{1}{2}$ back into the expression for $y$: $$ y = \frac{8\left(\frac{1}{2}\right) - 24}{5} $$ $$ y = \frac{4 - 24}{5} $$ $$ y = \frac{-20}{5} = -4 $$
The solution is: ( \left( \frac{1}{2}, -4 \right) )
More Information
The values ( x = \frac{1}{2} ) and ( y = -4 ) represent the point where the two lines intersect on a coordinate plane, indicating that these values satisfy both equations simultaneously.
Tips
- Incorrect substitution: Be careful when substituting the isolated variable. Make sure to apply the entire expression.
- Ignoring the signs: Pay attention to the signs when distributing or rearranging terms, as this can lead to errors in solving.
AI-generated content may contain errors. Please verify critical information
