Solve x² + y² = 25, x + y = −1
Worked out line by line the way a teacher would write it.
Answer
| Solutions | x = 3, y = −4; x = −4, y = 3 |
Step-by-step solution
14 steps-
1 The system, with the equations numbered\begin{aligned}x^{2} + y^{2} &= 25 & (1)\\ x + y &= -1 & (2)\end{aligned}
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2 Solve equation (2) for x
Substitution starts by solving one equation for one unknown, ideally one with coefficient 1 or −1. Then x can be replaced in the other equation.
x = - y - 1 -
3 Substitute into equation (1)
Replacing the unknown by its expression leaves one equation with one unknown, which can be solved on its own.
y^{2} + \left(- y - 1\right)^{2} = 25 -
4 Expand the brackets and collect like terms
Multiply out each bracket with the distributive property, a(b + c) = ab + ac, then combine like terms: terms with the same power of the variable, such as 3x and −x, add up to 2x.
2 y^{2} + 2 y + 1 = 25 -
5 Move every term to the left side so the right side is 0
Factoring and the quadratic formula both work on an equation of the form … = 0. Subtracting the right side from both sides gets there without changing the solutions.
2 y^{2} + 2 y - 24 = 0 -
6 Divide both sides by the common factor 2
Every coefficient is divisible by 2. Dividing both sides by 2 gives smaller numbers and the same solutions, because 0 divided by 2 is still 0.
y^{2} + y - 12 = 0 -
7 This is a quadratic in standard form ay² + by + c = 0
Every quadratic equation can be arranged as ay² + by + c = 0. Reading off a = 1, b = 1 and c = -12, signs included, shows which method fits: factoring, taking a square root or the quadratic formula.
a = 1,\quad b = 1,\quad c = -12 -
8 Factor the trinomial: find two numbers whose product is c = -12 and whose sum is b = 1
The goal is to write y² + by + c as (y + p)(y + q). Multiplying that out gives y² + (p + q)y + p·q, so p and q must multiply to c = -12 and add up to b = 1.
List the pairs of numbers whose product is -12, with their signs, and pick the pair whose sum is 1.
(-3) \cdot 4 = -12,\qquad (-3) + 4 = 1 -
9 Write the factored form\left(y - 3\right) \left(y + 4\right) = 0
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10 Zero product property: a product is 0 only when one of its factors is 0
0 is the only number with this property: if a·b = 0, then a = 0 or b = 0. That is why the equation was first rearranged to … = 0 and factored: now each factor can be set to 0 on its own, giving a simpler equation for each.
y - 3 = 0\quad \text{or} \quad y + 4 = 0 -
11 Add 3 to both sides
An equation stays true as long as you do the same thing to both sides, like adding the same weight to both pans of a balance. Adding or subtracting the same amount on both sides moves a term to the other side, where it appears with the opposite sign.
y = 3Move the constant terms to the right.
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12 Subtract 4 from both sidesy = -4
Move the constant terms to the right.
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13 Back-substitute y = -4
Once one unknown is known, putting its value into an earlier equation gives the other one.
x = -1 - -4 = 3 -
14 Back-substitute y = 3x = -1 - 3 = -4
Check by substitution
| Equation | Left side | Right side | |
|---|---|---|---|
| (1) | 25 | 25 | ✓ |
| (2) | −1 | −1 | ✓ |
| (1) | 25 | 25 | ✓ |
| (2) | −1 | −1 | ✓ |
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