Math  /  Geometry

QuestionSolve the following system by the addition method. x2+y2=25(x5)2+y2=10\begin{array}{l} x^{2}+y^{2}=25 \\ (x-5)^{2}+y^{2}=10 \end{array}
The solution set is \square \}. (Type an ordered pair. Use a comma to separate answers as needed. Type exact answers, using radicals as needed. Simplify your answer.)

Studdy Solution

STEP 1

What is this asking? We need to find the (x,y)(x, y) points where a circle and a shifted circle intersect! Watch out! Remember that when solving systems of equations, we're looking for values that satisfy *all* the equations at the same time.
Also, squaring can sometimes introduce extra solutions that don't actually work, so we'll need to **double-check** our answers!

STEP 2

1. Expand and Subtract
2. Solve for xx
3. Solve for yy
4. Verify Solutions

STEP 3

Let's **expand** (x5)2(x-5)^2 in the second equation: (x5)2+y2=x210x+25+y2=10 (x-5)^2 + y^2 = x^2 - 10x + 25 + y^2 = 10 So our second equation becomes: x210x+25+y2=10 x^2 - 10x + 25 + y^2 = 10

STEP 4

Notice how both equations have a y2y^2 term!
Let's **subtract** the second equation from the first equation to eliminate y2y^2: (x2+y2)(x210x+25+y2)=2510 (x^2 + y^2) - (x^2 - 10x + 25 + y^2) = 25 - 10 x2+y2x2+10x25y2=15 x^2 + y^2 - x^2 + 10x - 25 - y^2 = 15 10x25=15 10x - 25 = 15

STEP 5

We want to get xx by itself! **Add** 2525 to both sides of 10x25=1510x - 25 = 15: 10x25+25=15+25 10x - 25 + 25 = 15 + 25 10x=40 10x = 40

STEP 6

**Divide** both sides by 1010 to find our **xx value**: 10x10=4010 \frac{10x}{10} = \frac{40}{10} x=4 x = 4

STEP 7

Now that we know x=4x = 4, let's **substitute** it back into the first equation, x2+y2=25x^2 + y^2 = 25: (4)2+y2=25 (4)^2 + y^2 = 25 16+y2=25 16 + y^2 = 25

STEP 8

**Subtract** 1616 from both sides: 16+y216=2516 16 + y^2 - 16 = 25 - 16 y2=9 y^2 = 9

STEP 9

**Take the square root** of both sides to get yy: y2=9 \sqrt{y^2} = \sqrt{9} y=±3 y = \pm 3 So we have two possible yy values: y=3y = 3 and y=3y = -3.

STEP 10

We found two potential solutions: (4,3)(4, 3) and (4,3)(4, -3).
Let's make sure they work in *both* original equations.

STEP 11

For (4,3)(4, 3): (4)2+(3)2=16+9=25 (4)^2 + (3)^2 = 16 + 9 = 25 (45)2+(3)2=(1)2+9=1+9=10 (4 - 5)^2 + (3)^2 = (-1)^2 + 9 = 1 + 9 = 10 It works!

STEP 12

For (4,3)(4, -3): (4)2+(3)2=16+9=25 (4)^2 + (-3)^2 = 16 + 9 = 25 (45)2+(3)2=(1)2+9=1+9=10 (4 - 5)^2 + (-3)^2 = (-1)^2 + 9 = 1 + 9 = 10 It works too!

STEP 13

The solution set is {(4,3),(4,3)}\{(4, 3), (4, -3)\}.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord