Solve the System of Equations x^2+y^2=100 y=-1/7x+50/7

$x^{2} + y^{2} = 100$ $y = - \frac{1}{7} x + \frac{50}{7}$

Combine $x$ and $\frac{1}{7}$ .

$y = - \frac{x}{7} + \frac{50}{7}$

$x^{2} + y^{2} = 100$

Replace all occurrences of $y$ with $- \frac{x}{7} + \frac{50}{7}$ in each equation.

Replace all occurrences of $y$ in $x^{2} + y^{2} = 100$ with $- \frac{x}{7} + \frac{50}{7}$ .

$x^{2} + {(- \frac{x}{7} + \frac{50}{7})}^{2} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Simplify the left side.

Simplify $x^{2} + {(- \frac{x}{7} + \frac{50}{7})}^{2}$ .

Simplify each term.

Rewrite ${(- \frac{x}{7} + \frac{50}{7})}^{2}$ as $(- \frac{x}{7} + \frac{50}{7}) (- \frac{x}{7} + \frac{50}{7})$ .

$x^{2} + (- \frac{x}{7} + \frac{50}{7}) (- \frac{x}{7} + \frac{50}{7}) = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Expand $(- \frac{x}{7} + \frac{50}{7}) (- \frac{x}{7} + \frac{50}{7})$ using the FOIL Method.

Apply the distributive property.

$x^{2} - \frac{x}{7} \cdot (- \frac{x}{7} + \frac{50}{7}) + \frac{50}{7} \cdot (- \frac{x}{7} + \frac{50}{7}) = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Apply the distributive property.

$x^{2} - \frac{x}{7} \cdot (- \frac{x}{7}) - \frac{x}{7} \cdot \frac{50}{7} + \frac{50}{7} \cdot (- \frac{x}{7} + \frac{50}{7}) = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Apply the distributive property.

$x^{2} - \frac{x}{7} \cdot (- \frac{x}{7}) - \frac{x}{7} \cdot \frac{50}{7} + \frac{50}{7} \cdot (- \frac{x}{7}) + \frac{50}{7} \cdot \frac{50}{7} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

$x^{2} - \frac{x}{7} \cdot (- \frac{x}{7}) - \frac{x}{7} \cdot \frac{50}{7} + \frac{50}{7} \cdot (- \frac{x}{7}) + \frac{50}{7} \cdot \frac{50}{7} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Simplify and combine like terms.

Simplify each term.

Multiply $- \frac{x}{7} (- \frac{x}{7})$ .

Multiply $- 1$ by $- 1$ .

$x^{2} + 1 (\frac{x}{7}) \cdot \frac{x}{7} - \frac{x}{7} \cdot \frac{50}{7} + \frac{50}{7} \cdot (- \frac{x}{7}) + \frac{50}{7} \cdot \frac{50}{7} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Multiply $\frac{x}{7}$ by $1$ .

$x^{2} + \frac{x}{7} \cdot \frac{x}{7} - \frac{x}{7} \cdot \frac{50}{7} + \frac{50}{7} \cdot (- \frac{x}{7}) + \frac{50}{7} \cdot \frac{50}{7} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Multiply $\frac{x}{7}$ by $\frac{x}{7}$ .

$x^{2} + \frac{x \cdot x}{7 \cdot 7} - \frac{x}{7} \cdot \frac{50}{7} + \frac{50}{7} \cdot (- \frac{x}{7}) + \frac{50}{7} \cdot \frac{50}{7} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Raise $x$ to the power of $1$ .

$x^{2} + \frac{x \cdot x}{7 \cdot 7} - \frac{x}{7} \cdot \frac{50}{7} + \frac{50}{7} \cdot (- \frac{x}{7}) + \frac{50}{7} \cdot \frac{50}{7} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Raise $x$ to the power of $1$ .

$x^{2} + \frac{x \cdot x}{7 \cdot 7} - \frac{x}{7} \cdot \frac{50}{7} + \frac{50}{7} \cdot (- \frac{x}{7}) + \frac{50}{7} \cdot \frac{50}{7} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2} + \frac{x^{1 + 1}}{7 \cdot 7} - \frac{x}{7} \cdot \frac{50}{7} + \frac{50}{7} \cdot (- \frac{x}{7}) + \frac{50}{7} \cdot \frac{50}{7} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Add $1$ and $1$ .

$x^{2} + \frac{x^{2}}{7 \cdot 7} - \frac{x}{7} \cdot \frac{50}{7} + \frac{50}{7} \cdot (- \frac{x}{7}) + \frac{50}{7} \cdot \frac{50}{7} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Multiply $7$ by $7$ .

$x^{2} + \frac{x^{2}}{49} - \frac{x}{7} \cdot \frac{50}{7} + \frac{50}{7} \cdot (- \frac{x}{7}) + \frac{50}{7} \cdot \frac{50}{7} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

$x^{2} + \frac{x^{2}}{49} - \frac{x}{7} \cdot \frac{50}{7} + \frac{50}{7} \cdot (- \frac{x}{7}) + \frac{50}{7} \cdot \frac{50}{7} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Multiply $- \frac{x}{7} \cdot \frac{50}{7}$ .

Multiply $\frac{50}{7}$ by $\frac{x}{7}$ .

$x^{2} + \frac{x^{2}}{49} - \frac{50 x}{7 \cdot 7} + \frac{50}{7} \cdot (- \frac{x}{7}) + \frac{50}{7} \cdot \frac{50}{7} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Multiply $7$ by $7$ .

$x^{2} + \frac{x^{2}}{49} - \frac{50 x}{49} + \frac{50}{7} \cdot (- \frac{x}{7}) + \frac{50}{7} \cdot \frac{50}{7} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

$x^{2} + \frac{x^{2}}{49} - \frac{50 x}{49} + \frac{50}{7} \cdot (- \frac{x}{7}) + \frac{50}{7} \cdot \frac{50}{7} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Multiply $\frac{50}{7} (- \frac{x}{7})$ .

Multiply $\frac{50}{7}$ by $\frac{x}{7}$ .

$x^{2} + \frac{x^{2}}{49} - \frac{50 x}{49} - \frac{50 x}{7 \cdot 7} + \frac{50}{7} \cdot \frac{50}{7} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Multiply $7$ by $7$ .

$x^{2} + \frac{x^{2}}{49} - \frac{50 x}{49} - \frac{50 x}{49} + \frac{50}{7} \cdot \frac{50}{7} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

$x^{2} + \frac{x^{2}}{49} - \frac{50 x}{49} - \frac{50 x}{49} + \frac{50}{7} \cdot \frac{50}{7} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Multiply $\frac{50}{7} \cdot \frac{50}{7}$ .

Multiply $\frac{50}{7}$ by $\frac{50}{7}$ .

$x^{2} + \frac{x^{2}}{49} - \frac{50 x}{49} - \frac{50 x}{49} + \frac{50 \cdot 50}{7 \cdot 7} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Multiply $50$ by $50$ .

$x^{2} + \frac{x^{2}}{49} - \frac{50 x}{49} - \frac{50 x}{49} + \frac{2500}{7 \cdot 7} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Multiply $7$ by $7$ .

$x^{2} + \frac{x^{2}}{49} - \frac{50 x}{49} - \frac{50 x}{49} + \frac{2500}{49} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

$x^{2} + \frac{x^{2}}{49} - \frac{50 x}{49} - \frac{50 x}{49} + \frac{2500}{49} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

$x^{2} + \frac{x^{2}}{49} - \frac{50 x}{49} - \frac{50 x}{49} + \frac{2500}{49} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Subtract $\frac{50 x}{49}$ from $- \frac{50 x}{49}$ .

$x^{2} + \frac{x^{2}}{49} - 2 \frac{50 x}{49} + \frac{2500}{49} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

$x^{2} + \frac{x^{2}}{49} - 2 \frac{50 x}{49} + \frac{2500}{49} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Simplify each term.

Multiply $- 2 \frac{50 x}{49}$ .

Combine $- 2$ and $\frac{50 x}{49}$ .

$x^{2} + \frac{x^{2}}{49} + \frac{- 2 (50 x)}{49} + \frac{2500}{49} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Multiply $50$ by $- 2$ .

$x^{2} + \frac{x^{2}}{49} + \frac{- 100 x}{49} + \frac{2500}{49} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

$x^{2} + \frac{x^{2}}{49} + \frac{- 100 x}{49} + \frac{2500}{49} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Move the negative in front of the fraction.

$x^{2} + \frac{x^{2}}{49} - \frac{100 x}{49} + \frac{2500}{49} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

$x^{2} + \frac{x^{2}}{49} - \frac{100 x}{49} + \frac{2500}{49} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

$x^{2} + \frac{x^{2}}{49} - \frac{100 x}{49} + \frac{2500}{49} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

To write $x^{2}$ as a fraction with a common denominator, multiply by $\frac{49}{49}$ .

$x^{2} \cdot \frac{49}{49} + \frac{x^{2}}{49} - \frac{100 x}{49} + \frac{2500}{49} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Simplify terms.

Combine $x^{2}$ and $\frac{49}{49}$ .

$\frac{x^{2} \cdot 49}{49} + \frac{x^{2}}{49} - \frac{100 x}{49} + \frac{2500}{49} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 49 + x^{2}}{49} - \frac{100 x}{49} + \frac{2500}{49} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 49 + x^{2} - 100 x + 2500}{49} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

$\frac{x^{2} \cdot 49 + x^{2} - 100 x + 2500}{49} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Move $49$ to the left of $x^{2}$ .

$\frac{49 x^{2} + x^{2} - 100 x + 2500}{49} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Simplify terms.

Add $49 x^{2}$ and $x^{2}$ .

$\frac{50 x^{2} - 100 x + 2500}{49} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Factor $50$ out of $50 x^{2} - 100 x + 2500$ .

Factor $50$ out of $50 x^{2}$ .

$\frac{50 (x^{2}) - 100 x + 2500}{49} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Factor $50$ out of $- 100 x$ .

$\frac{50 (x^{2}) + 50 (- 2 x) + 2500}{49} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Factor $50$ out of $2500$ .

$\frac{50 x^{2} + 50 (- 2 x) + 50 \cdot 50}{49} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Factor $50$ out of $50 x^{2} + 50 (- 2 x)$ .

$\frac{50 (x^{2} - 2 x) + 50 \cdot 50}{49} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Factor $50$ out of $50 (x^{2} - 2 x) + 50 \cdot 50$ .

$\frac{50 (x^{2} - 2 x + 50)}{49} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

$\frac{50 (x^{2} - 2 x + 50)}{49} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

$\frac{50 (x^{2} - 2 x + 50)}{49} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

$\frac{50 (x^{2} - 2 x + 50)}{49} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

$\frac{50 (x^{2} - 2 x + 50)}{49} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

$\frac{50 (x^{2} - 2 x + 50)}{49} = 100$

$y = - \frac{x}{7} + \frac{50}{7}$

Solve for $x$ in $\frac{50 (x^{2} - 2 x + 50)}{49} = 100$ .

Multiply both sides by $49$ .

$\frac{50 (x^{2} - 2 x + 50)}{49} \cdot 49 = 100 \cdot 49$

$y = - \frac{x}{7} + \frac{50}{7}$

Simplify.

Simplify the left side.

Simplify $\frac{50 (x^{2} - 2 x + 50)}{49} \cdot 49$ .

Cancel the common factor of $49$ .

Cancel the common factor.

$\frac{50 (x^{2} - 2 x + 50)}{49} \cdot 49 = 100 \cdot 49$

$y = - \frac{x}{7} + \frac{50}{7}$

Rewrite the expression.

$50 (x^{2} - 2 x + 50) = 100 \cdot 49$

$y = - \frac{x}{7} + \frac{50}{7}$

$50 (x^{2} - 2 x + 50) = 100 \cdot 49$

$y = - \frac{x}{7} + \frac{50}{7}$

Apply the distributive property.

$50 x^{2} + 50 (- 2 x) + 50 \cdot 50 = 100 \cdot 49$

$y = - \frac{x}{7} + \frac{50}{7}$

Simplify.

Multiply $- 2$ by $50$ .

$50 x^{2} - 100 x + 50 \cdot 50 = 100 \cdot 49$

$y = - \frac{x}{7} + \frac{50}{7}$

Multiply $50$ by $50$ .

$50 x^{2} - 100 x + 2500 = 100 \cdot 49$

$y = - \frac{x}{7} + \frac{50}{7}$

$50 x^{2} - 100 x + 2500 = 100 \cdot 49$

$y = - \frac{x}{7} + \frac{50}{7}$

$50 x^{2} - 100 x + 2500 = 100 \cdot 49$

$y = - \frac{x}{7} + \frac{50}{7}$

$50 x^{2} - 100 x + 2500 = 100 \cdot 49$

$y = - \frac{x}{7} + \frac{50}{7}$

Simplify the right side.

Multiply $100$ by $49$ .

$50 x^{2} - 100 x + 2500 = 4900$

$y = - \frac{x}{7} + \frac{50}{7}$

$50 x^{2} - 100 x + 2500 = 4900$

$y = - \frac{x}{7} + \frac{50}{7}$

$50 x^{2} - 100 x + 2500 = 4900$

$y = - \frac{x}{7} + \frac{50}{7}$

Solve for $x$ .

Subtract $4900$ from both sides of the equation.

$50 x^{2} - 100 x + 2500 - 4900 = 0$

$y = - \frac{x}{7} + \frac{50}{7}$

Subtract $4900$ from $2500$ .

$50 x^{2} - 100 x - 2400 = 0$

$y = - \frac{x}{7} + \frac{50}{7}$

Factor the left side of the equation.

Factor $50$ out of $50 x^{2} - 100 x - 2400$ .

Factor $50$ out of $50 x^{2}$ .

$50 (x^{2}) - 100 x - 2400 = 0$

$y = - \frac{x}{7} + \frac{50}{7}$

Factor $50$ out of $- 100 x$ .

$50 (x^{2}) + 50 (- 2 x) - 2400 = 0$

$y = - \frac{x}{7} + \frac{50}{7}$

Factor $50$ out of $- 2400$ .

$50 x^{2} + 50 (- 2 x) + 50 \cdot - 48 = 0$

$y = - \frac{x}{7} + \frac{50}{7}$

Factor $50$ out of $50 x^{2} + 50 (- 2 x)$ .

$50 (x^{2} - 2 x) + 50 \cdot - 48 = 0$

$y = - \frac{x}{7} + \frac{50}{7}$

Factor $50$ out of $50 (x^{2} - 2 x) + 50 \cdot - 48$ .

$50 (x^{2} - 2 x - 48) = 0$

$y = - \frac{x}{7} + \frac{50}{7}$

$50 (x^{2} - 2 x - 48) = 0$

$y = - \frac{x}{7} + \frac{50}{7}$

Factor.

Factor $x^{2} - 2 x - 48$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 48$ and whose sum is $- 2$ .

$- 8, 6$

$y = - \frac{x}{7} + \frac{50}{7}$

Write the factored form using these integers.

$50 ((x - 8) (x + 6)) = 0$

$y = - \frac{x}{7} + \frac{50}{7}$

$50 ((x - 8) (x + 6)) = 0$

$y = - \frac{x}{7} + \frac{50}{7}$

Remove unnecessary parentheses.

$50 (x - 8) (x + 6) = 0$

$y = - \frac{x}{7} + \frac{50}{7}$

$50 (x - 8) (x + 6) = 0$

$y = - \frac{x}{7} + \frac{50}{7}$

$50 (x - 8) (x + 6) = 0$

$y = - \frac{x}{7} + \frac{50}{7}$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 8 = 0$

$x + 6 = 0$

$y = - \frac{x}{7} + \frac{50}{7}$

Set $x - 8$ equal to $0$ and solve for $x$ .

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

$y = - \frac{x}{7} + \frac{50}{7}$

Add $8$ to both sides of the equation.

$x = 8$

$y = - \frac{x}{7} + \frac{50}{7}$

$x = 8$

$y = - \frac{x}{7} + \frac{50}{7}$

Set $x + 6$ equal to $0$ and solve for $x$ .

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

$y = - \frac{x}{7} + \frac{50}{7}$

Subtract $6$ from both sides of the equation.

$x = - 6$

$y = - \frac{x}{7} + \frac{50}{7}$

$x = - 6$

$y = - \frac{x}{7} + \frac{50}{7}$

The final solution is all the values that make $50 (x - 8) (x + 6) = 0$ true.

$x = 8, - 6$

$y = - \frac{x}{7} + \frac{50}{7}$

$x = 8, - 6$

$y = - \frac{x}{7} + \frac{50}{7}$

$x = 8, - 6$

$y = - \frac{x}{7} + \frac{50}{7}$

Replace all occurrences of $x$ with $8$ in each equation.

Replace all occurrences of $x$ in $y = - \frac{x}{7} + \frac{50}{7}$ with $8$ .

$y = - \frac{8}{7} + \frac{50}{7}$

$x = 8$

Simplify the right side.

Simplify $- \frac{8}{7} + \frac{50}{7}$ .

Combine the numerators over the common denominator.

$y = \frac{- 8 + 50}{7}$

$x = 8$

Simplify the expression.

Add $- 8$ and $50$ .

$y = \frac{42}{7}$

$x = 8$

Divide $42$ by $7$ .

$y = 6$

$x = 8$

$y = 6$

$x = 8$

$y = 6$

$x = 8$

$y = 6$

$x = 8$

$y = 6$

$x = 8$

Replace all occurrences of $x$ with $- 6$ in each equation.

Replace all occurrences of $x$ in $y = - \frac{x}{7} + \frac{50}{7}$ with $- 6$ .

$y = - \frac{- 6}{7} + \frac{50}{7}$

$x = - 6$

Simplify the right side.

Simplify $- \frac{- 6}{7} + \frac{50}{7}$ .

Combine the numerators over the common denominator.

$y = \frac{6 + 50}{7}$

$x = - 6$

Simplify the expression.

Add $6$ and $50$ .

$y = \frac{56}{7}$

$x = - 6$

Divide $56$ by $7$ .

$y = 8$

$x = - 6$

$y = 8$

$x = - 6$

$y = 8$

$x = - 6$

$y = 8$

$x = - 6$

$y = 8$

$x = - 6$

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(8, 6)$

$(- 6, 8)$

The result can be shown in multiple forms.

Point Form:

$(8, 6), (- 6, 8)$

Equation Form:

$x = 8, y = 6$

$x = - 6, y = 8$

Solve the System of Equations x^2+y^2=100 y=-1/7x+50/7

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