Solve by Substitution 4a+2b=20 , 15a+5b=60

$4 a + 2 b = 20$ , $15 a + 5 b = 60$

Step 1

Solve for $a$ in $4 a + 2 b = 20$ .

Subtract $2 b$ from both sides of the equation.

$4 a = 20 - 2 b$

$15 a + 5 b = 60$

Divide each term in $4 a = 20 - 2 b$ by $4$ and simplify.

Divide each term in $4 a = 20 - 2 b$ by $4$ .

$\frac{4 a}{4} = \frac{20}{4} + \frac{- 2 b}{4}$

$15 a + 5 b = 60$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 a}{4} = \frac{20}{4} + \frac{- 2 b}{4}$

$15 a + 5 b = 60$

Divide $a$ by $1$ .

$a = \frac{20}{4} + \frac{- 2 b}{4}$

$15 a + 5 b = 60$

$a = \frac{20}{4} + \frac{- 2 b}{4}$

$15 a + 5 b = 60$

$a = \frac{20}{4} + \frac{- 2 b}{4}$

$15 a + 5 b = 60$

Simplify the right side.

Simplify each term.

Divide $20$ by $4$ .

$a = 5 + \frac{- 2 b}{4}$

$15 a + 5 b = 60$

Cancel the common factor of $- 2$ and $4$ .

Factor $2$ out of $- 2 b$ .

$a = 5 + \frac{2 (- b)}{4}$

$15 a + 5 b = 60$

Cancel the common factors.

Factor $2$ out of $4$ .

$a = 5 + \frac{2 (- b)}{2 (2)}$

$15 a + 5 b = 60$

Cancel the common factor.

$a = 5 + \frac{2 (- b)}{2 \cdot 2}$

$15 a + 5 b = 60$

Rewrite the expression.

$a = 5 + \frac{- b}{2}$

$15 a + 5 b = 60$

$a = 5 + \frac{- b}{2}$

$15 a + 5 b = 60$

$a = 5 + \frac{- b}{2}$

$15 a + 5 b = 60$

Move the negative in front of the fraction.

$a = 5 - \frac{b}{2}$

$15 a + 5 b = 60$

$a = 5 - \frac{b}{2}$

$15 a + 5 b = 60$

$a = 5 - \frac{b}{2}$

$15 a + 5 b = 60$

$a = 5 - \frac{b}{2}$

$15 a + 5 b = 60$

$a = 5 - \frac{b}{2}$

$15 a + 5 b = 60$

Step 2

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

Replace all occurrences of $a$ in $15 a + 5 b = 60$ with $5 - \frac{b}{2}$ .

$15 (5 - \frac{b}{2}) + 5 b = 60$

$a = 5 - \frac{b}{2}$

Simplify the left side.

Simplify $15 (5 - \frac{b}{2}) + 5 b$ .

Simplify each term.

Apply the distributive property.

$15 \cdot 5 + 15 (- \frac{b}{2}) + 5 b = 60$

$a = 5 - \frac{b}{2}$

Multiply $15$ by $5$ .

$75 + 15 (- \frac{b}{2}) + 5 b = 60$

$a = 5 - \frac{b}{2}$

Multiply $15 (- \frac{b}{2})$ .

Multiply $- 1$ by $15$ .

$75 - 15 \frac{b}{2} + 5 b = 60$

$a = 5 - \frac{b}{2}$

Combine $- 15$ and $\frac{b}{2}$ .

$75 + \frac{- 15 b}{2} + 5 b = 60$

$a = 5 - \frac{b}{2}$

$75 + \frac{- 15 b}{2} + 5 b = 60$

$a = 5 - \frac{b}{2}$

Move the negative in front of the fraction.

$75 - \frac{15 b}{2} + 5 b = 60$

$a = 5 - \frac{b}{2}$

$75 - \frac{15 b}{2} + 5 b = 60$

$a = 5 - \frac{b}{2}$

To write $5 b$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$75 - \frac{15 b}{2} + 5 b \cdot \frac{2}{2} = 60$

$a = 5 - \frac{b}{2}$

Simplify terms.

Combine $5 b$ and $\frac{2}{2}$ .

$75 - \frac{15 b}{2} + \frac{5 b \cdot 2}{2} = 60$

$a = 5 - \frac{b}{2}$

Combine the numerators over the common denominator.

$75 + \frac{- 15 b + 5 b \cdot 2}{2} = 60$

$a = 5 - \frac{b}{2}$

$75 + \frac{- 15 b + 5 b \cdot 2}{2} = 60$

$a = 5 - \frac{b}{2}$

Simplify each term.

Simplify the numerator.

Factor $5 b$ out of $- 15 b + 5 b \cdot 2$ .

Factor $5 b$ out of $- 15 b$ .

$75 + \frac{5 b (- 3) + 5 b \cdot 2}{2} = 60$

$a = 5 - \frac{b}{2}$

Factor $5 b$ out of $5 b \cdot 2$ .

$75 + \frac{5 b (- 3) + 5 b (2)}{2} = 60$

$a = 5 - \frac{b}{2}$

Factor $5 b$ out of $5 b (- 3) + 5 b (2)$ .

$75 + \frac{5 b (- 3 + 2)}{2} = 60$

$a = 5 - \frac{b}{2}$

$75 + \frac{5 b (- 3 + 2)}{2} = 60$

$a = 5 - \frac{b}{2}$

Add $- 3$ and $2$ .

$75 + \frac{5 b \cdot - 1}{2} = 60$

$a = 5 - \frac{b}{2}$

Multiply $- 1$ by $5$ .

$75 + \frac{- 5 b}{2} = 60$

$a = 5 - \frac{b}{2}$

$75 + \frac{- 5 b}{2} = 60$

$a = 5 - \frac{b}{2}$

Move the negative in front of the fraction.

$75 - \frac{5 b}{2} = 60$

$a = 5 - \frac{b}{2}$

$75 - \frac{5 b}{2} = 60$

$a = 5 - \frac{b}{2}$

$75 - \frac{5 b}{2} = 60$

$a = 5 - \frac{b}{2}$

$75 - \frac{5 b}{2} = 60$

$a = 5 - \frac{b}{2}$

$75 - \frac{5 b}{2} = 60$

$a = 5 - \frac{b}{2}$

Step 3

Solve for $b$ in $75 - \frac{5 b}{2} = 60$ .

Move all terms not containing $b$ to the right side of the equation.

Subtract $75$ from both sides of the equation.

$- \frac{5 b}{2} = 60 - 75$

$a = 5 - \frac{b}{2}$

Subtract $75$ from $60$ .

$- \frac{5 b}{2} = - 15$

$a = 5 - \frac{b}{2}$

$- \frac{5 b}{2} = - 15$

$a = 5 - \frac{b}{2}$

Multiply both sides of the equation by $- \frac{2}{5}$ .

$- \frac{2}{5} \cdot (- \frac{5 b}{2}) = - \frac{2}{5} \cdot - 15$

$a = 5 - \frac{b}{2}$

Simplify both sides of the equation.

Simplify the left side.

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

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{2}{5}$ into the numerator.

$\frac{- 2}{5} \cdot (- \frac{5 b}{2}) = - \frac{2}{5} \cdot - 15$

$a = 5 - \frac{b}{2}$

Move the leading negative in $- \frac{5 b}{2}$ into the numerator.

$\frac{- 2}{5} \cdot \frac{- 5 b}{2} = - \frac{2}{5} \cdot - 15$

$a = 5 - \frac{b}{2}$

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{5} \cdot \frac{- 5 b}{2} = - \frac{2}{5} \cdot - 15$

$a = 5 - \frac{b}{2}$

Cancel the common factor.

$\frac{2 \cdot - 1}{5} \cdot \frac{- 5 b}{2} = - \frac{2}{5} \cdot - 15$

$a = 5 - \frac{b}{2}$

Rewrite the expression.

$\frac{- 1}{5} \cdot (- 5 b) = - \frac{2}{5} \cdot - 15$

$a = 5 - \frac{b}{2}$

$\frac{- 1}{5} \cdot (- 5 b) = - \frac{2}{5} \cdot - 15$

$a = 5 - \frac{b}{2}$

Cancel the common factor of $5$ .

Factor $5$ out of $- 5 b$ .

$\frac{- 1}{5} \cdot (5 (- b)) = - \frac{2}{5} \cdot - 15$

$a = 5 - \frac{b}{2}$

Cancel the common factor.

$\frac{- 1}{5} \cdot (5 (- b)) = - \frac{2}{5} \cdot - 15$

$a = 5 - \frac{b}{2}$

Rewrite the expression.

$b = - \frac{2}{5} \cdot - 15$

$a = 5 - \frac{b}{2}$

$b = - \frac{2}{5} \cdot - 15$

$a = 5 - \frac{b}{2}$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 b = - \frac{2}{5} \cdot - 15$

$a = 5 - \frac{b}{2}$

Multiply $b$ by $1$ .

$b = - \frac{2}{5} \cdot - 15$

$a = 5 - \frac{b}{2}$

$b = - \frac{2}{5} \cdot - 15$

$a = 5 - \frac{b}{2}$

$b = - \frac{2}{5} \cdot - 15$

$a = 5 - \frac{b}{2}$

$b = - \frac{2}{5} \cdot - 15$

$a = 5 - \frac{b}{2}$

Simplify the right side.

Simplify $- \frac{2}{5} \cdot - 15$ .

Cancel the common factor of $5$ .

Move the leading negative in $- \frac{2}{5}$ into the numerator.

$b = \frac{- 2}{5} \cdot - 15$

$a = 5 - \frac{b}{2}$

Factor $5$ out of $- 15$ .

$b = \frac{- 2}{5} \cdot (5 (- 3))$

$a = 5 - \frac{b}{2}$

Cancel the common factor.

$b = \frac{- 2}{5} \cdot (5 \cdot - 3)$

$a = 5 - \frac{b}{2}$

Rewrite the expression.

$b = - 2 \cdot - 3$

$a = 5 - \frac{b}{2}$

$b = - 2 \cdot - 3$

$a = 5 - \frac{b}{2}$

Multiply $- 2$ by $- 3$ .

$b = 6$

$a = 5 - \frac{b}{2}$

$b = 6$

$a = 5 - \frac{b}{2}$

$b = 6$

$a = 5 - \frac{b}{2}$

$b = 6$

$a = 5 - \frac{b}{2}$

$b = 6$

$a = 5 - \frac{b}{2}$

Step 4

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

Replace all occurrences of $b$ in $a = 5 - \frac{b}{2}$ with $6$ .

$a = 5 - \frac{6}{2}$

$b = 6$

Simplify the right side.

Simplify $5 - \frac{6}{2}$ .

Simplify each term.

Divide $6$ by $2$ .

$a = 5 - 1 \cdot 3$

$b = 6$

Multiply $- 1$ by $3$ .

$a = 5 - 3$

$b = 6$

$a = 5 - 3$

$b = 6$

Subtract $3$ from $5$ .

$a = 2$

$b = 6$

$a = 2$

$b = 6$

$a = 2$

$b = 6$

$a = 2$

$b = 6$

Step 5

List all of the solutions.

$a = 2, b = 6$

Solve by Substitution 4a+2b=20 , 15a+5b=60

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