Solve by Substitution 25x^2+36y^2=900 , 6y+5x=30

$25 x^{2} + 36 y^{2} = 900$ , $6 y + 5 x = 30$

Step 1

Solve for $y$ in $6 y + 5 x = 30$ .

Subtract $5 x$ from both sides of the equation.

$6 y = 30 - 5 x$

$25 x^{2} + 36 y^{2} = 900$

Divide each term in $6 y = 30 - 5 x$ by $6$ and simplify.

Divide each term in $6 y = 30 - 5 x$ by $6$ .

$\frac{6 y}{6} = \frac{30}{6} + \frac{- 5 x}{6}$

$25 x^{2} + 36 y^{2} = 900$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 y}{6} = \frac{30}{6} + \frac{- 5 x}{6}$

$25 x^{2} + 36 y^{2} = 900$

Divide $y$ by $1$ .

$y = \frac{30}{6} + \frac{- 5 x}{6}$

$25 x^{2} + 36 y^{2} = 900$

$y = \frac{30}{6} + \frac{- 5 x}{6}$

$25 x^{2} + 36 y^{2} = 900$

$y = \frac{30}{6} + \frac{- 5 x}{6}$

$25 x^{2} + 36 y^{2} = 900$

Simplify the right side.

Simplify each term.

Divide $30$ by $6$ .

$y = 5 + \frac{- 5 x}{6}$

$25 x^{2} + 36 y^{2} = 900$

Move the negative in front of the fraction.

$y = 5 - \frac{5 x}{6}$

$25 x^{2} + 36 y^{2} = 900$

$y = 5 - \frac{5 x}{6}$

$25 x^{2} + 36 y^{2} = 900$

$y = 5 - \frac{5 x}{6}$

$25 x^{2} + 36 y^{2} = 900$

$y = 5 - \frac{5 x}{6}$

$25 x^{2} + 36 y^{2} = 900$

$y = 5 - \frac{5 x}{6}$

$25 x^{2} + 36 y^{2} = 900$

Step 2

Replace all occurrences of $y$ with $5 - \frac{5 x}{6}$ in each equation.

Replace all occurrences of $y$ in $25 x^{2} + 36 y^{2} = 900$ with $5 - \frac{5 x}{6}$ .

$25 x^{2} + 36 {(5 - \frac{5 x}{6})}^{2} = 900$

$y = 5 - \frac{5 x}{6}$

Simplify the left side.

Simplify $25 x^{2} + 36 {(5 - \frac{5 x}{6})}^{2}$ .

Simplify each term.

Rewrite ${(5 - \frac{5 x}{6})}^{2}$ as $(5 - \frac{5 x}{6}) (5 - \frac{5 x}{6})$ .

$25 x^{2} + 36 ((5 - \frac{5 x}{6}) (5 - \frac{5 x}{6})) = 900$

$y = 5 - \frac{5 x}{6}$

Expand $(5 - \frac{5 x}{6}) (5 - \frac{5 x}{6})$ using the FOIL Method.

Apply the distributive property.

$25 x^{2} + 36 (5 (5 - \frac{5 x}{6}) - \frac{5 x}{6} \cdot (5 - \frac{5 x}{6})) = 900$

$y = 5 - \frac{5 x}{6}$

Apply the distributive property.

$25 x^{2} + 36 (5 \cdot 5 + 5 (- \frac{5 x}{6}) - \frac{5 x}{6} \cdot (5 - \frac{5 x}{6})) = 900$

$y = 5 - \frac{5 x}{6}$

Apply the distributive property.

$25 x^{2} + 36 (5 \cdot 5 + 5 (- \frac{5 x}{6}) - \frac{5 x}{6} \cdot 5 - \frac{5 x}{6} \cdot (- \frac{5 x}{6})) = 900$

$y = 5 - \frac{5 x}{6}$

$25 x^{2} + 36 (5 \cdot 5 + 5 (- \frac{5 x}{6}) - \frac{5 x}{6} \cdot 5 - \frac{5 x}{6} \cdot (- \frac{5 x}{6})) = 900$

$y = 5 - \frac{5 x}{6}$

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $5$ .

$25 x^{2} + 36 (25 + 5 (- \frac{5 x}{6}) - \frac{5 x}{6} \cdot 5 - \frac{5 x}{6} \cdot (- \frac{5 x}{6})) = 900$

$y = 5 - \frac{5 x}{6}$

Multiply $5 (- \frac{5 x}{6})$ .

Multiply $- 1$ by $5$ .

$25 x^{2} + 36 (25 - 5 \frac{5 x}{6} - \frac{5 x}{6} \cdot 5 - \frac{5 x}{6} \cdot (- \frac{5 x}{6})) = 900$

$y = 5 - \frac{5 x}{6}$

Combine $- 5$ and $\frac{5 x}{6}$ .

$25 x^{2} + 36 (25 + \frac{- 5 (5 x)}{6} - \frac{5 x}{6} \cdot 5 - \frac{5 x}{6} \cdot (- \frac{5 x}{6})) = 900$

$y = 5 - \frac{5 x}{6}$

Multiply $5$ by $- 5$ .

$25 x^{2} + 36 (25 + \frac{- 25 x}{6} - \frac{5 x}{6} \cdot 5 - \frac{5 x}{6} \cdot (- \frac{5 x}{6})) = 900$

$y = 5 - \frac{5 x}{6}$

$25 x^{2} + 36 (25 + \frac{- 25 x}{6} - \frac{5 x}{6} \cdot 5 - \frac{5 x}{6} \cdot (- \frac{5 x}{6})) = 900$

$y = 5 - \frac{5 x}{6}$

Move the negative in front of the fraction.

$25 x^{2} + 36 (25 - \frac{(25) x}{6} - \frac{5 x}{6} \cdot 5 - \frac{5 x}{6} \cdot (- \frac{5 x}{6})) = 900$

$y = 5 - \frac{5 x}{6}$

Multiply $- \frac{5 x}{6} \cdot 5$ .

Multiply $5$ by $- 1$ .

$25 x^{2} + 36 (25 - \frac{25 x}{6} - 5 \frac{5 x}{6} - \frac{5 x}{6} \cdot (- \frac{5 x}{6})) = 900$

$y = 5 - \frac{5 x}{6}$

Combine $- 5$ and $\frac{5 x}{6}$ .

$25 x^{2} + 36 (25 - \frac{25 x}{6} + \frac{- 5 (5 x)}{6} - \frac{5 x}{6} \cdot (- \frac{5 x}{6})) = 900$

$y = 5 - \frac{5 x}{6}$

Multiply $5$ by $- 5$ .

$25 x^{2} + 36 (25 - \frac{25 x}{6} + \frac{- 25 x}{6} - \frac{5 x}{6} \cdot (- \frac{5 x}{6})) = 900$

$y = 5 - \frac{5 x}{6}$

$25 x^{2} + 36 (25 - \frac{25 x}{6} + \frac{- 25 x}{6} - \frac{5 x}{6} \cdot (- \frac{5 x}{6})) = 900$

$y = 5 - \frac{5 x}{6}$

Move the negative in front of the fraction.

$25 x^{2} + 36 (25 - \frac{25 x}{6} - \frac{(25) x}{6} - \frac{5 x}{6} \cdot (- \frac{5 x}{6})) = 900$

$y = 5 - \frac{5 x}{6}$

Multiply $- \frac{5 x}{6} (- \frac{5 x}{6})$ .

Multiply $- 1$ by $- 1$ .

$25 x^{2} + 36 (25 - \frac{25 x}{6} - \frac{25 x}{6} + 1 (\frac{5 x}{6}) \cdot \frac{5 x}{6}) = 900$

$y = 5 - \frac{5 x}{6}$

Multiply $\frac{5 x}{6}$ by $1$ .

$25 x^{2} + 36 (25 - \frac{25 x}{6} - \frac{25 x}{6} + \frac{5 x}{6} \cdot \frac{5 x}{6}) = 900$

$y = 5 - \frac{5 x}{6}$

Multiply $\frac{5 x}{6}$ by $\frac{5 x}{6}$ .

$25 x^{2} + 36 (25 - \frac{25 x}{6} - \frac{25 x}{6} + \frac{5 x (5 x)}{6 \cdot 6}) = 900$

$y = 5 - \frac{5 x}{6}$

Multiply $5$ by $5$ .

$25 x^{2} + 36 (25 - \frac{25 x}{6} - \frac{25 x}{6} + \frac{25 x \cdot x}{6 \cdot 6}) = 900$

$y = 5 - \frac{5 x}{6}$

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

$25 x^{2} + 36 (25 - \frac{25 x}{6} - \frac{25 x}{6} + \frac{25 (x \cdot x)}{6 \cdot 6}) = 900$

$y = 5 - \frac{5 x}{6}$

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

$25 x^{2} + 36 (25 - \frac{25 x}{6} - \frac{25 x}{6} + \frac{25 (x \cdot x)}{6 \cdot 6}) = 900$

$y = 5 - \frac{5 x}{6}$

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

$25 x^{2} + 36 (25 - \frac{25 x}{6} - \frac{25 x}{6} + \frac{25 x^{1 + 1}}{6 \cdot 6}) = 900$

$y = 5 - \frac{5 x}{6}$

Add $1$ and $1$ .

$25 x^{2} + 36 (25 - \frac{25 x}{6} - \frac{25 x}{6} + \frac{25 x^{2}}{6 \cdot 6}) = 900$

$y = 5 - \frac{5 x}{6}$

Multiply $6$ by $6$ .

$25 x^{2} + 36 (25 - \frac{25 x}{6} - \frac{25 x}{6} + \frac{25 x^{2}}{36}) = 900$

$y = 5 - \frac{5 x}{6}$

$25 x^{2} + 36 (25 - \frac{25 x}{6} - \frac{25 x}{6} + \frac{25 x^{2}}{36}) = 900$

$y = 5 - \frac{5 x}{6}$

$25 x^{2} + 36 (25 - \frac{25 x}{6} - \frac{25 x}{6} + \frac{25 x^{2}}{36}) = 900$

$y = 5 - \frac{5 x}{6}$

Subtract $\frac{25 x}{6}$ from $- \frac{25 x}{6}$ .

$25 x^{2} + 36 (25 - 2 \frac{25 x}{6} + \frac{25 x^{2}}{36}) = 900$

$y = 5 - \frac{5 x}{6}$

$25 x^{2} + 36 (25 - 2 \frac{25 x}{6} + \frac{25 x^{2}}{36}) = 900$

$y = 5 - \frac{5 x}{6}$

Simplify each term.

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

$25 x^{2} + 36 (25 + 2 (- 1) (\frac{25 x}{6}) + \frac{25 x^{2}}{36}) = 900$

$y = 5 - \frac{5 x}{6}$

Factor $2$ out of $6$ .

$25 x^{2} + 36 (25 + 2 \cdot (- 1 \frac{25 x}{2 \cdot 3}) + \frac{25 x^{2}}{36}) = 900$

$y = 5 - \frac{5 x}{6}$

Cancel the common factor.

$25 x^{2} + 36 (25 + 2 \cdot (- 1 \frac{25 x}{2 \cdot 3}) + \frac{25 x^{2}}{36}) = 900$

$y = 5 - \frac{5 x}{6}$

Rewrite the expression.

$25 x^{2} + 36 (25 - 1 \frac{25 x}{3} + \frac{25 x^{2}}{36}) = 900$

$y = 5 - \frac{5 x}{6}$

$25 x^{2} + 36 (25 - 1 \frac{25 x}{3} + \frac{25 x^{2}}{36}) = 900$

$y = 5 - \frac{5 x}{6}$

Rewrite $- 1 \frac{25 x}{3}$ as $- \frac{25 x}{3}$ .

$25 x^{2} + 36 (25 - \frac{25 x}{3} + \frac{25 x^{2}}{36}) = 900$

$y = 5 - \frac{5 x}{6}$

$25 x^{2} + 36 (25 - \frac{25 x}{3} + \frac{25 x^{2}}{36}) = 900$

$y = 5 - \frac{5 x}{6}$

Apply the distributive property.

$25 x^{2} + 36 \cdot 25 + 36 (- \frac{25 x}{3}) + 36 (\frac{25 x^{2}}{36}) = 900$

$y = 5 - \frac{5 x}{6}$

Simplify.

Multiply $36$ by $25$ .

$25 x^{2} + 900 + 36 (- \frac{25 x}{3}) + 36 (\frac{25 x^{2}}{36}) = 900$

$y = 5 - \frac{5 x}{6}$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{25 x}{3}$ into the numerator.

$25 x^{2} + 900 + 36 (\frac{- 25 x}{3}) + 36 (\frac{25 x^{2}}{36}) = 900$

$y = 5 - \frac{5 x}{6}$

Factor $3$ out of $36$ .

$25 x^{2} + 900 + 3 (12) (\frac{- 25 x}{3}) + 36 (\frac{25 x^{2}}{36}) = 900$

$y = 5 - \frac{5 x}{6}$

Cancel the common factor.

$25 x^{2} + 900 + 3 \cdot (12 (\frac{- 25 x}{3})) + 36 (\frac{25 x^{2}}{36}) = 900$

$y = 5 - \frac{5 x}{6}$

Rewrite the expression.

$25 x^{2} + 900 + 12 (- 25 x) + 36 (\frac{25 x^{2}}{36}) = 900$

$y = 5 - \frac{5 x}{6}$

$25 x^{2} + 900 + 12 (- 25 x) + 36 (\frac{25 x^{2}}{36}) = 900$

$y = 5 - \frac{5 x}{6}$

Multiply $- 25$ by $12$ .

$25 x^{2} + 900 - 300 x + 36 (\frac{25 x^{2}}{36}) = 900$

$y = 5 - \frac{5 x}{6}$

Cancel the common factor of $36$ .

Cancel the common factor.

$25 x^{2} + 900 - 300 x + 36 (\frac{25 x^{2}}{36}) = 900$

$y = 5 - \frac{5 x}{6}$

Rewrite the expression.

$25 x^{2} + 900 - 300 x + 25 x^{2} = 900$

$y = 5 - \frac{5 x}{6}$

$25 x^{2} + 900 - 300 x + 25 x^{2} = 900$

$y = 5 - \frac{5 x}{6}$

$25 x^{2} + 900 - 300 x + 25 x^{2} = 900$

$y = 5 - \frac{5 x}{6}$

$25 x^{2} + 900 - 300 x + 25 x^{2} = 900$

$y = 5 - \frac{5 x}{6}$

Add $25 x^{2}$ and $25 x^{2}$ .

$50 x^{2} + 900 - 300 x = 900$

$y = 5 - \frac{5 x}{6}$

$50 x^{2} + 900 - 300 x = 900$

$y = 5 - \frac{5 x}{6}$

$50 x^{2} + 900 - 300 x = 900$

$y = 5 - \frac{5 x}{6}$

$50 x^{2} + 900 - 300 x = 900$

$y = 5 - \frac{5 x}{6}$

Step 3

Solve for $x$ in $50 x^{2} + 900 - 300 x = 900$ .

Subtract $900$ from both sides of the equation.

$50 x^{2} + 900 - 300 x - 900 = 0$

$y = 5 - \frac{5 x}{6}$

Combine the opposite terms in $50 x^{2} + 900 - 300 x - 900$ .

Subtract $900$ from $900$ .

$50 x^{2} - 300 x + 0 = 0$

$y = 5 - \frac{5 x}{6}$

Add $50 x^{2} - 300 x$ and $0$ .

$50 x^{2} - 300 x = 0$

$y = 5 - \frac{5 x}{6}$

$50 x^{2} - 300 x = 0$

$y = 5 - \frac{5 x}{6}$

Factor $50 x$ out of $50 x^{2} - 300 x$ .

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

$50 x (x) - 300 x = 0$

$y = 5 - \frac{5 x}{6}$

Factor $50 x$ out of $- 300 x$ .

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

$y = 5 - \frac{5 x}{6}$

Factor $50 x$ out of $50 x (x) + 50 x (- 6)$ .

$50 x (x - 6) = 0$

$y = 5 - \frac{5 x}{6}$

$50 x (x - 6) = 0$

$y = 5 - \frac{5 x}{6}$

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

$x = 0$

$x - 6 = 0$

$y = 5 - \frac{5 x}{6}$

Set $x$ equal to $0$ .

$x = 0$

$y = 5 - \frac{5 x}{6}$

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

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

$x - 6 = 0$

$y = 5 - \frac{5 x}{6}$

Add $6$ to both sides of the equation.

$x = 6$

$y = 5 - \frac{5 x}{6}$

$x = 6$

$y = 5 - \frac{5 x}{6}$

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

$x = 0, 6$

$y = 5 - \frac{5 x}{6}$

$x = 0, 6$

$y = 5 - \frac{5 x}{6}$

Step 4

Replace all occurrences of $x$ with $0$ in each equation.

Replace all occurrences of $x$ in $y = 5 - \frac{5 x}{6}$ with $0$ .

$y = 5 - \frac{5 (0)}{6}$

$x = 0$

Simplify the right side.

Simplify $5 - \frac{5 (0)}{6}$ .

Simplify each term.

Cancel the common factor of $0$ and $6$ .

Factor $6$ out of $5 (0)$ .

$y = 5 - \frac{6 (5 \cdot (0))}{6}$

$x = 0$

Cancel the common factors.

Factor $6$ out of $6$ .

$y = 5 - \frac{6 (5 \cdot (0))}{6 (1)}$

$x = 0$

Cancel the common factor.

$y = 5 - \frac{6 (5 \cdot (0))}{6 \cdot 1}$

$x = 0$

Rewrite the expression.

$y = 5 - \frac{5 \cdot (0)}{1}$

$x = 0$

Divide $5 \cdot (0)$ by $1$ .

$y = 5 - (5 \cdot (0))$

$x = 0$

$y = 5 - (5 \cdot (0))$

$x = 0$

$y = 5 - (5 \cdot (0))$

$x = 0$

Multiply $- (5 \cdot (0))$ .

Multiply $5$ by $0$ .

$y = 5 - 0$

$x = 0$

Multiply $- 1$ by $0$ .

$y = 5 + 0$

$x = 0$

$y = 5 + 0$

$x = 0$

$y = 5 + 0$

$x = 0$

Add $5$ and $0$ .

$y = 5$

$x = 0$

$y = 5$

$x = 0$

$y = 5$

$x = 0$

$y = 5$

$x = 0$

Step 5

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

Replace all occurrences of $x$ in $y = 5 - \frac{5 x}{6}$ with $6$ .

$y = 5 - \frac{5 (6)}{6}$

$x = 6$

Simplify the right side.

Simplify $5 - \frac{5 (6)}{6}$ .

Simplify each term.

Cancel the common factor of $6$ .

Cancel the common factor.

$y = 5 - \frac{5 \cdot 6}{6}$

$x = 6$

Divide $5$ by $1$ .

$y = 5 - 1 \cdot 5$

$x = 6$

$y = 5 - 1 \cdot 5$

$x = 6$

Multiply $- 1$ by $5$ .

$y = 5 - 5$

$x = 6$

$y = 5 - 5$

$x = 6$

Subtract $5$ from $5$ .

$y = 0$

$x = 6$

$y = 0$

$x = 6$

$y = 0$

$x = 6$

$y = 0$

$x = 6$

Step 6

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

$(0, 5)$

$(6, 0)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(0, 5), (6, 0)$

Equation Form:

$x = 0, y = 5$

$x = 6, y = 0$

Step 8

Solve by Substitution 25x^2+36y^2=900 , 6y+5x=30

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