Finite Math Examples

$a + 10 = 2 (3 - 10)$ , $\frac{3}{4} \cdot (a + 10) = b + 10$

Step 1

Move variables to the left and constant terms to the right.

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Step 1.1

Subtract $10$ from $3$ .

$a + 10 = 2 \cdot - 7$

$\frac{3}{4} \cdot (a + 10) = b + 10$

Step 1.2

Multiply $2$ by $- 7$ .

$a + 10 = - 14$

$\frac{3}{4} \cdot (a + 10) = b + 10$

Step 1.3

Move all terms not containing a variable to the right side of the equation.

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Step 1.3.1

Subtract $10$ from both sides of the equation.

$a = - 14 - 10$

$\frac{3}{4} \cdot (a + 10) = b + 10$

Step 1.3.2

Subtract $10$ from $- 14$ .

$a = - 24$

$\frac{3}{4} \cdot (a + 10) = b + 10$

$a = - 24$

$\frac{3}{4} \cdot (a + 10) = b + 10$

Step 1.4

Subtract $b$ from both sides of the equation.

$a = - 24$

$\frac{3}{4} \cdot (a + 10) - b = 10$

Step 1.5

Simplify each term.

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Step 1.5.1

Apply the distributive property.

$a = - 24$

$\frac{3}{4} a + \frac{3}{4} \cdot 10 - b = 10$

Step 1.5.2

Combine $\frac{3}{4}$ and $a$ .

$a = - 24$

$\frac{3 a}{4} + \frac{3}{4} \cdot 10 - b = 10$

Step 1.5.3

Cancel the common factor of $2$ .

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Step 1.5.3.1

Factor $2$ out of $4$ .

$a = - 24$

$\frac{3 a}{4} + \frac{3}{2 (2)} \cdot 10 - b = 10$

Step 1.5.3.2

Factor $2$ out of $10$ .

$a = - 24$

$\frac{3 a}{4} + \frac{3}{2 \cdot 2} \cdot (2 \cdot 5) - b = 10$

Step 1.5.3.3

Cancel the common factor.

$a = - 24$

$\frac{3 a}{4} + \frac{3}{2 \cdot 2} \cdot (2 \cdot 5) - b = 10$

Step 1.5.3.4

Rewrite the expression.

$a = - 24$

$\frac{3 a}{4} + \frac{3}{2} \cdot 5 - b = 10$

$a = - 24$

$\frac{3 a}{4} + \frac{3}{2} \cdot 5 - b = 10$

Step 1.5.4

Combine $\frac{3}{2}$ and $5$ .

$a = - 24$

$\frac{3 a}{4} + \frac{3 \cdot 5}{2} - b = 10$

Step 1.5.5

Multiply $3$ by $5$ .

$a = - 24$

$\frac{3 a}{4} + \frac{15}{2} - b = 10$

$a = - 24$

$\frac{3 a}{4} + \frac{15}{2} - b = 10$

Step 1.6

Move all terms not containing a variable to the right side of the equation.

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Step 1.6.1

Subtract $\frac{15}{2}$ from both sides of the equation.

$a = - 24$

$\frac{3 a}{4} - b = 10 - \frac{15}{2}$

Step 1.6.2

To write $10$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$a = - 24$

$\frac{3 a}{4} - b = 10 \cdot \frac{2}{2} - \frac{15}{2}$

Step 1.6.3

Combine $10$ and $\frac{2}{2}$ .

$a = - 24$

$\frac{3 a}{4} - b = \frac{10 \cdot 2}{2} - \frac{15}{2}$

Step 1.6.4

Combine the numerators over the common denominator.

$a = - 24$

$\frac{3 a}{4} - b = \frac{10 \cdot 2 - 15}{2}$

Step 1.6.5

Simplify the numerator.

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Step 1.6.5.1

Multiply $10$ by $2$ .

$a = - 24$

$\frac{3 a}{4} - b = \frac{20 - 15}{2}$

Step 1.6.5.2

Subtract $15$ from $20$ .

$a = - 24$

$\frac{3 a}{4} - b = \frac{5}{2}$

$a = - 24$

$\frac{3 a}{4} - b = \frac{5}{2}$

$a = - 24$

$\frac{3 a}{4} - b = \frac{5}{2}$

Step 1.7

Reorder terms.

$a = - 24$

$\frac{3}{4} a - b = \frac{5}{2}$

$a = - 24$

$\frac{3}{4} a - b = \frac{5}{2}$

Step 2

Write the system as a matrix.

$[\begin{array}{c} 1 & 0 & - 24 \\ \frac{3}{4} & - 1 & \frac{5}{2} \end{array}]$

Step 3

Find the reduced row echelon form.

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Step 3.1

Perform the row operation $R_{2} = R_{2} - \frac{3}{4} R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 3.1.1

Perform the row operation $R_{2} = R_{2} - \frac{3}{4} R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 0 & - 24 \\ \frac{3}{4} - \frac{3}{4} \cdot 1 & - 1 - \frac{3}{4} \cdot 0 & \frac{5}{2} - \frac{3}{4} \cdot - 24 \end{array}]$

Step 3.1.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0 & - 24 \\ 0 & - 1 & \frac{41}{2} \end{array}]$

Step 3.2

Multiply each element of $R_{2}$ by $- 1$ to make the entry at $2, 2$ a $1$ .

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Step 3.2.1

Multiply each element of $R_{2}$ by $- 1$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 0 & - 24 \\ - 0 & - - 1 & - \frac{41}{2} \end{array}]$

Step 3.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0 & - 24 \\ 0 & 1 & - \frac{41}{2} \end{array}]$

Step 4

Use the result matrix to declare the final solution to the system of equations.

$a = - 24$

$b = - \frac{41}{2}$

Step 5

The solution is the set of ordered pairs that make the system true.

$(- 24, - \frac{41}{2})$