Question

Solve the system of equations and check your solution.
x^2+y^2=25
x+2y=10

I tried but I got only the 2 x values 8 plus or minus the square root of 164 all over 2.
I know the answer is wrong but I can't find my mistake.
Can you show me step-by-step how to solve this system of equation?

Answer 1

Just a fun fact: the first equation is a circle with radius of 5 (square root of 25).

I'll use substitution to answer this.
Let's start with defining x: x=10-2*y

Insert into the first equation: (10-2*y)^2+y^2=25
Expand the equation: 100-20y-20y+4y^2+y^2=25
Simplify: 5y^2-40y+100=25
Simplify: 5y^2-40y+75=0
Divide both sides by 5:y^2-8y+15=0
Solve for 0 using the algebraic formula:
(8+(sqrt(8^2-4*1*15)))/2*1
=(8+sqrt(4))/2; sqrt(4)=2
and (8-sqrt(4))/2
So the answers are 10/2 and 6/2 or y=3 and y=5

Plug these back into the second equation:
x+2y=10
x+2*3=10 and x+2*5=10
x+6=10 and x+10=10
x=4 and x=0

You probably just made a multiplication error inside the square root.

You can check the answer by graphing the two and seeing where they intersect. I use Desmos for graphing.

Answer 2

x + 2 y = 10 Subtract x to both sides

x + 2 y - x = 10 - x

2 y = 10 - x Divide both sides by 2

2 y / 2 = 10 / 2 - x / 2

y = 5 - x / 2


x ^ 2 + y ^ 2 = 25

x ^ 2 + ( 5 - x / 2 ) ^ 2 = 25

________________________________
Remark:

( a - b ) ^ 2 = a ^ 2 - 2 * a * b + b ^ 2
_______________________________

x ^ 2 + [ 5 ^ 2 - 2 * 5 * x / 2 + ( x / 2 ) ^ 2 ] = 25

x ^ 2 + 25 - 5 x + x ^ 2 / 4 = 25

x ^ 2 + x ^ 2 / 4 + 25 - 5 x = 25

4 x ^ 2 / 4 + x ^ 2 / 4 + 25 - 5 x = 25

5 x ^ 2 / 4 + x ^ 2 + 25 - 5 x = 25 Subtract 25 to both sides

5 x ^ 2 / 4 + 25 - 5 x - 25 = 25 - 25

5 x ^ 2 / 4 - 5 x = 0 Divide both sides by 5

x ^ 2 / 4 - x = 0

x ( x / 4 - 1 ) = 0

Split into two equations.

x = 0

and

x / 4 - 1 = 0 Add 1 to both sides

x / 4 - 1 + 1 = 0 + 1

x / 4 = 1 Multiply both sides by

x = 4

Replace x = 0 and x = 4 into equation x + 2 y = 10

x = 0

0 + 2 y = 10

2 y = 10 Divide both sides by 2

y = 5


x = 4

4 + 2 y = 10 Subtract 4 to both sides

4 + 2 y - 4 = 10 - 4

2 y = 6 Divide both sides by 2

y = 3


The solutions are :

x = 0 , y = 5

and

x = 4 , y = 3

You can write solutions like :

( 0 , 5 ) and ( 4 , 3 )


Chek solutions :

x = 0 , y = 5

x ^ 2 + y ^ 2 = 25

0 ^ 2 + 5 ^ 2 = 25

0 + 25 = 25

25 = 25 Correct

x + 2 y = 10

0 + 2 * 5 = 10

0 + 10 = 10

10 = 10 Correct


x = 4 , y = 3

x ^ 2 + y ^ 2 = 25

4 ^ 2 + 3 ^ 2 = 25

16 + 9 = 25

25 = 25 Correct

x + 2 y = 10

4 + 2 * 3 = 10

4 + 6 = 10

10 = 10 Correct