Solve the following system by the elimination method.

Question

one fifth x plus y
equalssix fifths
one tenth x plus one third y
equalsseven tenths

Answer 1

To solve the system using the elimination method, let's first rewrite the equations based on your input:

1. $\frac{1}{5}x+y=\frac{6}{5}$
2. $\frac{1}{10}x+\frac{1}{3}y=\frac{7}{10}$

Next, we'll eliminate the fractions by multiplying both equations by the least common multiple (LCM) of the denominators.

Step 1: Eliminate Fractions

For the first equation, the LCM of 5 is 5. Multiply the entire equation by 5:

$$5(\frac{1}{5}x+y)=5\left(\frac{6}{5}\right)$$

This simplifies to:

$$x+5y=6\text{(Equation 1)}$$

For the second equation, the LCM of 10 and 3 is 30. Multiply the entire equation by 30:

$$30(\frac{1}{10}x+\frac{1}{3}y)=30\left(\frac{7}{10}\right)$$

This simplifies to:

$$3x+10y=21\text{(Equation 2)}$$

Step 2: Set Up the New System

Now we have the following system of equations:

1. $x+5y=6$
2. $3x+10y=21$

Step 3: Elimination Method

We can multiply Equation 1 by 3 to eliminate $x$:

$$3(x+5y)=3(6)$$ which gives: $$3x+15y=18\text{(Equation 3)}$$

Now we have the modified system:

1. $3x+15y=18$ (from Equation 1)
2. $3x+10y=21$ (Equation 2)

Step 4: Subtract Equations

Subtract Equation 2 from Equation 3:

$$(3x+15y)-(3x+10y)=18-21$$

This simplifies to:

$$5y=-3$$

Step 5: Solve for $y$

Divide both sides by 5:

$$y=-\frac{3}{5}$$

Step 6: Substitute $y$ Back to Find $x$

Now substitute $y$ back into one of the original equations. We'll use Equation 1:

$$x+5(-\frac{3}{5})=6$$

This simplifies to:

$$x-3=6$$

Adding 3 to both sides gives:

$$x=9$$

Step 7: Final Solution

The solution to the system is:

$$x=9,y=-\frac{3}{5}$$

Thus, the final answer is:

$$(x,y)=(9,-\frac{3}{5})$$