Write a system of equations where the sum of two numbers is 10 and the difference is 6. Can you determine the solution without graphing? Guess and check.
Let's call the two numbers $x$ and $y$. We are given that the sum of the two numbers is 10, so we can write the equation $x+y=10$. We are also given that the difference between the two numbers is 6, so we can write the equation $x-y=6$.
We can solve this system of equations by adding the two equations together to eliminate $y$. Adding the left sides gives $x+y$, and adding the right sides gives $10+6=16$. So our equation is $x+y=16$.
If we subtract the second equation from the first equation, we can eliminate $y$ again. Subtracting the left sides gives $x-y$, and subtracting the right sides gives $6-10=-4$. So our equation is $x-y=-4$.
We have now reduced the system of equations to a single equation: $x+y=16$ and $x-y=-4$. To solve for $x$ and $y$, we can add the two equations together to eliminate $y$: $(x+y)+(x-y)=16+(-4)$. Simplifying the left side gives $2x$, and simplifying the right side gives $12$. So our equation is $2x=12$.
Dividing both sides of the equation by 2 gives $x=\frac{12}{2}=6$. Plugging this value of $x$ into either of the original equations, we find $y=10-x=10-6=4$.
So the solution to the system of equations is $x=6$ and $y=4$.
How is that correct if the equation x-y=6
Has to = 6
6-4=2 NOT 6