solve the system of equations algebraically for x and y
{ x^2 + y^2 = 26
x + y = 6
a. (1,5) (1,4)
b. (4,5) (5,4)
c. (5,1) (5,1)
d. (1,5) (5,1)
e. no solution
Given : x^2+y^2=26
x+y=6
squaring on both sides
(x+y)^2=6^2
x^2+y^2+2xy=36
==>26+2xy=36
==>2xy=10
==>xy=5
==>x=5/y
now (5/y)^2+y^2=26
==>y^4-26y^2+25=0
on futher solving
x=1 or 5
y= 5 or 1
To solve the system of equations algebraically, we can use the substitution method. Here are the steps:
Step 1: Using the second equation, solve for x in terms of y.
x + y = 6
x = 6 - y
Step 2: Substitute the value of x from Step 1 into the first equation.
(6 - y)^2 + y^2 = 26
Step 3: Expand and simplify the equation.
36 - 12y + y^2 + y^2 = 26
2y^2 - 12y + 10 = 0
Step 4: Divide the equation by 2 to simplify.
y^2 - 6y + 5 = 0
Step 5: Factor the quadratic equation.
(y - 1)(y - 5) = 0
Step 6: Set each factor equal to zero and solve for y.
y - 1 = 0 or y - 5 = 0
y = 1 or y = 5
Step 7: Substitute the values of y into the equation x = 6 - y to solve for x.
When y = 1, x = 6 - 1 = 5
When y = 5, x = 6 - 5 = 1
Therefore, the solution to the system of equations is (x, y) = (5, 1) or (1, 5).
The correct answer is: d. (5, 1) (1, 5).
To solve the system of equations algebraically for x and y, we can use the method of substitution or elimination. Let's use the method of substitution:
1. Start with the second equation: x + y = 6.
2. Solve this equation for one variable, preferably x or y. Let's solve it for x:
x = 6 - y.
3. Substitute the value of x into the first equation: (6 - y)^2 + y^2 = 26.
4. Expand and simplify the equation: 36 - 12y + y^2 + y^2 = 26.
Combine like terms: 2y^2 - 12y + 10 = 0.
5. Divide the entire equation by 2 to simplify it: y^2 - 6y + 5 = 0.
6. Factor the quadratic equation: (y - 5)(y - 1) = 0.
7. Set each factor to zero and solve for y:
y - 5 = 0 or y - 1 = 0.
y = 5 or y = 1.
8. Substitute the values of y back into x = 6 - y to solve for x:
If y = 5, then x = 6 - 5 = 1.
If y = 1, then x = 6 - 1 = 5.
So the solution to the system of equations x^2 + y^2 = 26 and x + y = 6 is (1, 5) or (5, 1).
Therefore, the correct answer is (d) (1, 5) and (5, 1).