Asked by Anonymous
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?
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?
Answers
Answered by
Kent
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.
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.
Answered by
Bosnian
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
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
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