Solve by Substitution 25x^2+4y^2=100 , 2y+5x=10

$25 x^{2} + 4 y^{2} = 100$ , $2 y + 5 x = 10$

Step 1

Solve for $y$ in $2 y + 5 x = 10$ .

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

$2 y = 10 - 5 x$

$25 x^{2} + 4 y^{2} = 100$

Divide each term in $2 y = 10 - 5 x$ by $2$ and simplify.

Divide each term in $2 y = 10 - 5 x$ by $2$ .

$\frac{2 y}{2} = \frac{10}{2} + \frac{- 5 x}{2}$

$25 x^{2} + 4 y^{2} = 100$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 y}{2} = \frac{10}{2} + \frac{- 5 x}{2}$

$25 x^{2} + 4 y^{2} = 100$

Divide $y$ by $1$ .

$y = \frac{10}{2} + \frac{- 5 x}{2}$

$25 x^{2} + 4 y^{2} = 100$

$y = \frac{10}{2} + \frac{- 5 x}{2}$

$25 x^{2} + 4 y^{2} = 100$

$y = \frac{10}{2} + \frac{- 5 x}{2}$

$25 x^{2} + 4 y^{2} = 100$

Simplify the right side.

Simplify each term.

Divide $10$ by $2$ .

$y = 5 + \frac{- 5 x}{2}$

$25 x^{2} + 4 y^{2} = 100$

Move the negative in front of the fraction.

$y = 5 - \frac{5 x}{2}$

$25 x^{2} + 4 y^{2} = 100$

$y = 5 - \frac{5 x}{2}$

$25 x^{2} + 4 y^{2} = 100$

$y = 5 - \frac{5 x}{2}$

$25 x^{2} + 4 y^{2} = 100$

$y = 5 - \frac{5 x}{2}$

$25 x^{2} + 4 y^{2} = 100$

$y = 5 - \frac{5 x}{2}$

$25 x^{2} + 4 y^{2} = 100$

Step 2

Replace all occurrences of $y$ with $5 - \frac{5 x}{2}$ in each equation.

Replace all occurrences of $y$ in $25 x^{2} + 4 y^{2} = 100$ with $5 - \frac{5 x}{2}$ .

$25 x^{2} + 4 {(5 - \frac{5 x}{2})}^{2} = 100$

$y = 5 - \frac{5 x}{2}$

Simplify the left side.

Simplify $25 x^{2} + 4 {(5 - \frac{5 x}{2})}^{2}$ .

Simplify each term.

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

$25 x^{2} + 4 ((5 - \frac{5 x}{2}) (5 - \frac{5 x}{2})) = 100$

$y = 5 - \frac{5 x}{2}$

Expand $(5 - \frac{5 x}{2}) (5 - \frac{5 x}{2})$ using the FOIL Method.

Apply the distributive property.

$25 x^{2} + 4 (5 (5 - \frac{5 x}{2}) - \frac{5 x}{2} \cdot (5 - \frac{5 x}{2})) = 100$

$y = 5 - \frac{5 x}{2}$

Apply the distributive property.

$25 x^{2} + 4 (5 \cdot 5 + 5 (- \frac{5 x}{2}) - \frac{5 x}{2} \cdot (5 - \frac{5 x}{2})) = 100$

$y = 5 - \frac{5 x}{2}$

Apply the distributive property.

$25 x^{2} + 4 (5 \cdot 5 + 5 (- \frac{5 x}{2}) - \frac{5 x}{2} \cdot 5 - \frac{5 x}{2} \cdot (- \frac{5 x}{2})) = 100$

$y = 5 - \frac{5 x}{2}$

$25 x^{2} + 4 (5 \cdot 5 + 5 (- \frac{5 x}{2}) - \frac{5 x}{2} \cdot 5 - \frac{5 x}{2} \cdot (- \frac{5 x}{2})) = 100$

$y = 5 - \frac{5 x}{2}$

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $5$ .

$25 x^{2} + 4 (25 + 5 (- \frac{5 x}{2}) - \frac{5 x}{2} \cdot 5 - \frac{5 x}{2} \cdot (- \frac{5 x}{2})) = 100$

$y = 5 - \frac{5 x}{2}$

Multiply $5 (- \frac{5 x}{2})$ .

Multiply $- 1$ by $5$ .

$25 x^{2} + 4 (25 - 5 \frac{5 x}{2} - \frac{5 x}{2} \cdot 5 - \frac{5 x}{2} \cdot (- \frac{5 x}{2})) = 100$

$y = 5 - \frac{5 x}{2}$

Combine $- 5$ and $\frac{5 x}{2}$ .

$25 x^{2} + 4 (25 + \frac{- 5 (5 x)}{2} - \frac{5 x}{2} \cdot 5 - \frac{5 x}{2} \cdot (- \frac{5 x}{2})) = 100$

$y = 5 - \frac{5 x}{2}$

Multiply $5$ by $- 5$ .

$25 x^{2} + 4 (25 + \frac{- 25 x}{2} - \frac{5 x}{2} \cdot 5 - \frac{5 x}{2} \cdot (- \frac{5 x}{2})) = 100$

$y = 5 - \frac{5 x}{2}$

$25 x^{2} + 4 (25 + \frac{- 25 x}{2} - \frac{5 x}{2} \cdot 5 - \frac{5 x}{2} \cdot (- \frac{5 x}{2})) = 100$

$y = 5 - \frac{5 x}{2}$

Move the negative in front of the fraction.

$25 x^{2} + 4 (25 - \frac{(25) x}{2} - \frac{5 x}{2} \cdot 5 - \frac{5 x}{2} \cdot (- \frac{5 x}{2})) = 100$

$y = 5 - \frac{5 x}{2}$

Multiply $- \frac{5 x}{2} \cdot 5$ .

Multiply $5$ by $- 1$ .

$25 x^{2} + 4 (25 - \frac{25 x}{2} - 5 \frac{5 x}{2} - \frac{5 x}{2} \cdot (- \frac{5 x}{2})) = 100$

$y = 5 - \frac{5 x}{2}$

Combine $- 5$ and $\frac{5 x}{2}$ .

$25 x^{2} + 4 (25 - \frac{25 x}{2} + \frac{- 5 (5 x)}{2} - \frac{5 x}{2} \cdot (- \frac{5 x}{2})) = 100$

$y = 5 - \frac{5 x}{2}$

Multiply $5$ by $- 5$ .

$25 x^{2} + 4 (25 - \frac{25 x}{2} + \frac{- 25 x}{2} - \frac{5 x}{2} \cdot (- \frac{5 x}{2})) = 100$

$y = 5 - \frac{5 x}{2}$

$25 x^{2} + 4 (25 - \frac{25 x}{2} + \frac{- 25 x}{2} - \frac{5 x}{2} \cdot (- \frac{5 x}{2})) = 100$

$y = 5 - \frac{5 x}{2}$

Move the negative in front of the fraction.

$25 x^{2} + 4 (25 - \frac{25 x}{2} - \frac{(25) x}{2} - \frac{5 x}{2} \cdot (- \frac{5 x}{2})) = 100$

$y = 5 - \frac{5 x}{2}$

Multiply $- \frac{5 x}{2} (- \frac{5 x}{2})$ .

Multiply $- 1$ by $- 1$ .

$25 x^{2} + 4 (25 - \frac{25 x}{2} - \frac{25 x}{2} + 1 (\frac{5 x}{2}) \cdot \frac{5 x}{2}) = 100$

$y = 5 - \frac{5 x}{2}$

Multiply $\frac{5 x}{2}$ by $1$ .

$25 x^{2} + 4 (25 - \frac{25 x}{2} - \frac{25 x}{2} + \frac{5 x}{2} \cdot \frac{5 x}{2}) = 100$

$y = 5 - \frac{5 x}{2}$

Multiply $\frac{5 x}{2}$ by $\frac{5 x}{2}$ .

$25 x^{2} + 4 (25 - \frac{25 x}{2} - \frac{25 x}{2} + \frac{5 x (5 x)}{2 \cdot 2}) = 100$

$y = 5 - \frac{5 x}{2}$

Multiply $5$ by $5$ .

$25 x^{2} + 4 (25 - \frac{25 x}{2} - \frac{25 x}{2} + \frac{25 x \cdot x}{2 \cdot 2}) = 100$

$y = 5 - \frac{5 x}{2}$

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

$25 x^{2} + 4 (25 - \frac{25 x}{2} - \frac{25 x}{2} + \frac{25 (x \cdot x)}{2 \cdot 2}) = 100$

$y = 5 - \frac{5 x}{2}$

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

$25 x^{2} + 4 (25 - \frac{25 x}{2} - \frac{25 x}{2} + \frac{25 (x \cdot x)}{2 \cdot 2}) = 100$

$y = 5 - \frac{5 x}{2}$

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

$25 x^{2} + 4 (25 - \frac{25 x}{2} - \frac{25 x}{2} + \frac{25 x^{1 + 1}}{2 \cdot 2}) = 100$

$y = 5 - \frac{5 x}{2}$

Add $1$ and $1$ .

$25 x^{2} + 4 (25 - \frac{25 x}{2} - \frac{25 x}{2} + \frac{25 x^{2}}{2 \cdot 2}) = 100$

$y = 5 - \frac{5 x}{2}$

Multiply $2$ by $2$ .

$25 x^{2} + 4 (25 - \frac{25 x}{2} - \frac{25 x}{2} + \frac{25 x^{2}}{4}) = 100$

$y = 5 - \frac{5 x}{2}$

$25 x^{2} + 4 (25 - \frac{25 x}{2} - \frac{25 x}{2} + \frac{25 x^{2}}{4}) = 100$

$y = 5 - \frac{5 x}{2}$

$25 x^{2} + 4 (25 - \frac{25 x}{2} - \frac{25 x}{2} + \frac{25 x^{2}}{4}) = 100$

$y = 5 - \frac{5 x}{2}$

Subtract $\frac{25 x}{2}$ from $- \frac{25 x}{2}$ .

$25 x^{2} + 4 (25 - 2 \frac{25 x}{2} + \frac{25 x^{2}}{4}) = 100$

$y = 5 - \frac{5 x}{2}$

$25 x^{2} + 4 (25 - 2 \frac{25 x}{2} + \frac{25 x^{2}}{4}) = 100$

$y = 5 - \frac{5 x}{2}$

Simplify each term.

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

$25 x^{2} + 4 (25 + 2 (- 1) (\frac{25 x}{2}) + \frac{25 x^{2}}{4}) = 100$

$y = 5 - \frac{5 x}{2}$

Cancel the common factor.

$25 x^{2} + 4 (25 + 2 \cdot (- 1 \frac{25 x}{2}) + \frac{25 x^{2}}{4}) = 100$

$y = 5 - \frac{5 x}{2}$

Rewrite the expression.

$25 x^{2} + 4 (25 - 1 (25 x) + \frac{25 x^{2}}{4}) = 100$

$y = 5 - \frac{5 x}{2}$

$25 x^{2} + 4 (25 - 1 (25 x) + \frac{25 x^{2}}{4}) = 100$

$y = 5 - \frac{5 x}{2}$

Multiply $25$ by $- 1$ .

$25 x^{2} + 4 (25 - 25 x + \frac{25 x^{2}}{4}) = 100$

$y = 5 - \frac{5 x}{2}$

$25 x^{2} + 4 (25 - 25 x + \frac{25 x^{2}}{4}) = 100$

$y = 5 - \frac{5 x}{2}$

Apply the distributive property.

$25 x^{2} + 4 \cdot 25 + 4 (- 25 x) + 4 (\frac{25 x^{2}}{4}) = 100$

$y = 5 - \frac{5 x}{2}$

Simplify.

Multiply $4$ by $25$ .

$25 x^{2} + 100 + 4 (- 25 x) + 4 (\frac{25 x^{2}}{4}) = 100$

$y = 5 - \frac{5 x}{2}$

Multiply $- 25$ by $4$ .

$25 x^{2} + 100 - 100 x + 4 (\frac{25 x^{2}}{4}) = 100$

$y = 5 - \frac{5 x}{2}$

Cancel the common factor of $4$ .

Cancel the common factor.

$25 x^{2} + 100 - 100 x + 4 (\frac{25 x^{2}}{4}) = 100$

$y = 5 - \frac{5 x}{2}$

Rewrite the expression.

$25 x^{2} + 100 - 100 x + 25 x^{2} = 100$

$y = 5 - \frac{5 x}{2}$

$25 x^{2} + 100 - 100 x + 25 x^{2} = 100$

$y = 5 - \frac{5 x}{2}$

$25 x^{2} + 100 - 100 x + 25 x^{2} = 100$

$y = 5 - \frac{5 x}{2}$

$25 x^{2} + 100 - 100 x + 25 x^{2} = 100$

$y = 5 - \frac{5 x}{2}$

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

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

$y = 5 - \frac{5 x}{2}$

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

$y = 5 - \frac{5 x}{2}$

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

$y = 5 - \frac{5 x}{2}$

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

$y = 5 - \frac{5 x}{2}$

Step 3

Solve for $x$ in $50 x^{2} + 100 - 100 x = 100$ .

Subtract $100$ from both sides of the equation.

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

$y = 5 - \frac{5 x}{2}$

Combine the opposite terms in $50 x^{2} + 100 - 100 x - 100$ .

Subtract $100$ from $100$ .

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

$y = 5 - \frac{5 x}{2}$

Add $50 x^{2} - 100 x$ and $0$ .

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

$y = 5 - \frac{5 x}{2}$

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

$y = 5 - \frac{5 x}{2}$

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

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

$50 x (x) - 100 x = 0$

$y = 5 - \frac{5 x}{2}$

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

$50 x (x) + 50 x (- 2) = 0$

$y = 5 - \frac{5 x}{2}$

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

$50 x (x - 2) = 0$

$y = 5 - \frac{5 x}{2}$

$50 x (x - 2) = 0$

$y = 5 - \frac{5 x}{2}$

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

$x = 0$

$x - 2 = 0$

$y = 5 - \frac{5 x}{2}$

Set $x$ equal to $0$ .

$x = 0$

$y = 5 - \frac{5 x}{2}$

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

$y = 5 - \frac{5 x}{2}$

Add $2$ to both sides of the equation.

$x = 2$

$y = 5 - \frac{5 x}{2}$

$x = 2$

$y = 5 - \frac{5 x}{2}$

The final solution is all the values that make $50 x (x - 2) = 0$ true.

$x = 0, 2$

$y = 5 - \frac{5 x}{2}$

$x = 0, 2$

$y = 5 - \frac{5 x}{2}$

Step 4

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

Replace all occurrences of $x$ in $y = 5 - \frac{5 x}{2}$ with $0$ .

$y = 5 - \frac{5 (0)}{2}$

$x = 0$

Simplify the right side.

Simplify $5 - \frac{5 (0)}{2}$ .

Simplify each term.

Cancel the common factor of $0$ and $2$ .

Factor $2$ out of $5 (0)$ .

$y = 5 - \frac{2 (5 \cdot (0))}{2}$

$x = 0$

Cancel the common factors.

Factor $2$ out of $2$ .

$y = 5 - \frac{2 (5 \cdot (0))}{2 (1)}$

$x = 0$

Cancel the common factor.

$y = 5 - \frac{2 (5 \cdot (0))}{2 \cdot 1}$

$x = 0$

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 $2$ in each equation.

Replace all occurrences of $x$ in $y = 5 - \frac{5 x}{2}$ with $2$ .

$y = 5 - \frac{5 (2)}{2}$

$x = 2$

Simplify the right side.

Simplify $5 - \frac{5 (2)}{2}$ .

Simplify each term.

Cancel the common factor of $2$ .

Cancel the common factor.

$y = 5 - \frac{5 \cdot 2}{2}$

$x = 2$

Divide $5$ by $1$ .

$y = 5 - 1 \cdot 5$

$x = 2$

$y = 5 - 1 \cdot 5$

$x = 2$

Multiply $- 1$ by $5$ .

$y = 5 - 5$

$x = 2$

$y = 5 - 5$

$x = 2$

Subtract $5$ from $5$ .

$y = 0$

$x = 2$

$y = 0$

$x = 2$

$y = 0$

$x = 2$

$y = 0$

$x = 2$

Step 6

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

$(0, 5)$

$(2, 0)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(0, 5), (2, 0)$

Equation Form:

$x = 0, y = 5$

$x = 2, y = 0$

Step 8

Solve by Substitution 25x^2+4y^2=100 , 2y+5x=10

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