Solve the system of linear equations by substitution: 1/3 x + y = -1; 1/3 x + 8y = 13.

Understand the Problem

The question is asking us to solve a system of linear equations using the substitution method. We need to isolate one variable in one equation and substitute it into the other equation.

Answer

The solution is $ x = -9 $ and $ y = 2 $.

Steps to Solve

Isolate one variable in the first equation

From the first equation, ( \frac{1}{3} x + y = -1 ), we can isolate ( y ):

$$ y = -1 - \frac{1}{3} x $$

Substitute the isolated variable into the second equation

Now we substitute ( y ) into the second equation, ( \frac{1}{3} x + 8y = 13 ):

$$ \frac{1}{3} x + 8\left(-1 - \frac{1}{3} x\right) = 13 $$

Solve for ( x )

Distributing the ( 8 ):

$$ \frac{1}{3} x - 8 - \frac{8}{3} x = 13 $$

Combine like terms:

$$ \left(\frac{1}{3} - \frac{8}{3}\right) x - 8 = 13 $$

This simplifies to:

$$ -\frac{7}{3} x - 8 = 13 $$

Add ( 8 ) to both sides:

$$ -\frac{7}{3} x = 21 $$

Now, multiply both sides by ( -\frac{3}{7} ):

$$ x = -\frac{3 \cdot 21}{7} = -9 $$

Substitute ( x ) back to find ( y )

Now substitute ( x = -9 ) back into the equation we derived for ( y ):

$$ y = -1 - \frac{1}{3}(-9) $$

$$ y = -1 + 3 = 2 $$

Final values

The solution to the system is:

$$ x = -9 $$ and $$ y = 2 $$

The solution to the system of equations is ( x = -9 ) and ( y = 2 ).

More Information

This system of linear equations represents two lines in a coordinate plane, and the solution provides the point where these lines intersect.

Tips

Confusing signs when distributing the negative or multiplying fractions. Ensure to carefully handle negative signs.
Failing to combine like terms correctly, leading to incorrect simplifications.

AI-generated content may contain errors. Please verify critical information

Thank you for voting!