Finite Math Examples

$- x + y = 2$ , $x + 2 y = 10$ , $3 x + y = 15$

Step 1

Add $x$ to both sides of the equation.

$y = 2 + x$

$x + 2 y = 10$

$3 x + y = 15$

Step 2

Replace all occurrences of $y$ with $2 + x$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $x + 2 y = 10$ with $2 + x$ .

$x + 2 (2 + x) = 10$

$y = 2 + x$

$3 x + y = 15$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $x + 2 (2 + x)$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$x + 2 \cdot 2 + 2 x = 10$

$y = 2 + x$

$3 x + y = 15$

Step 2.2.1.1.2

Multiply $2$ by $2$ .

$x + 4 + 2 x = 10$

$y = 2 + x$

$3 x + y = 15$

$x + 4 + 2 x = 10$

$y = 2 + x$

$3 x + y = 15$

Step 2.2.1.2

Add $x$ and $2 x$ .

$3 x + 4 = 10$

$y = 2 + x$

$3 x + y = 15$

$3 x + 4 = 10$

$y = 2 + x$

$3 x + y = 15$

$3 x + 4 = 10$

$y = 2 + x$

$3 x + y = 15$

Step 2.3

Replace all occurrences of $y$ in $3 x + y = 15$ with $2 + x$ .

$3 x + 2 + x = 15$

$3 x + 4 = 10$

$y = 2 + x$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $3 x + 2 + x$ .

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Step 2.4.1.1

Remove parentheses.

$3 x + 2 + x = 15$

$3 x + 4 = 10$

$y = 2 + x$

Step 2.4.1.2

Add $3 x$ and $x$ .

$4 x + 2 = 15$

$3 x + 4 = 10$

$y = 2 + x$

$4 x + 2 = 15$

$3 x + 4 = 10$

$y = 2 + x$

$4 x + 2 = 15$

$3 x + 4 = 10$

$y = 2 + x$

$4 x + 2 = 15$

$3 x + 4 = 10$

$y = 2 + x$

Step 3

Solve for $x$ in $4 x + 2 = 15$ .

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Step 3.1

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

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Step 3.1.1

Subtract $2$ from both sides of the equation.

$4 x = 15 - 2$

$3 x + 4 = 10$

$y = 2 + x$

Step 3.1.2

Subtract $2$ from $15$ .

$4 x = 13$

$3 x + 4 = 10$

$y = 2 + x$

$4 x = 13$

$3 x + 4 = 10$

$y = 2 + x$

Step 3.2

Divide each term in $4 x = 13$ by $4$ and simplify.

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Step 3.2.1

Divide each term in $4 x = 13$ by $4$ .

$\frac{4 x}{4} = \frac{13}{4}$

$3 x + 4 = 10$

$y = 2 + x$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $4$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{13}{4}$

$3 x + 4 = 10$

$y = 2 + x$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{13}{4}$

$3 x + 4 = 10$

$y = 2 + x$

$x = \frac{13}{4}$

$3 x + 4 = 10$

$y = 2 + x$

$x = \frac{13}{4}$

$3 x + 4 = 10$

$y = 2 + x$

$x = \frac{13}{4}$

$3 x + 4 = 10$

$y = 2 + x$

$x = \frac{13}{4}$

$3 x + 4 = 10$

$y = 2 + x$

Step 4

Replace all occurrences of $x$ with $\frac{13}{4}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $3 x + 4 = 10$ with $\frac{13}{4}$ .

$3 (\frac{13}{4}) + 4 = 10$

$x = \frac{13}{4}$

$y = 2 + x$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $3 (\frac{13}{4}) + 4$ .

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Step 4.2.1.1

Multiply $3 (\frac{13}{4})$ .

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Step 4.2.1.1.1

Combine $3$ and $\frac{13}{4}$ .

$\frac{3 \cdot 13}{4} + 4 = 10$

$x = \frac{13}{4}$

$y = 2 + x$

Step 4.2.1.1.2

Multiply $3$ by $13$ .

$\frac{39}{4} + 4 = 10$

$x = \frac{13}{4}$

$y = 2 + x$

$\frac{39}{4} + 4 = 10$

$x = \frac{13}{4}$

$y = 2 + x$

Step 4.2.1.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{39}{4} + 4 \cdot \frac{4}{4} = 10$

$x = \frac{13}{4}$

$y = 2 + x$

Step 4.2.1.3

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

$\frac{39}{4} + \frac{4 \cdot 4}{4} = 10$

$x = \frac{13}{4}$

$y = 2 + x$

Step 4.2.1.4

Combine the numerators over the common denominator.

$\frac{39 + 4 \cdot 4}{4} = 10$

$x = \frac{13}{4}$

$y = 2 + x$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $4$ by $4$ .

$\frac{39 + 16}{4} = 10$

$x = \frac{13}{4}$

$y = 2 + x$

Step 4.2.1.5.2

Add $39$ and $16$ .

$\frac{55}{4} = 10$

$x = \frac{13}{4}$

$y = 2 + x$

$\frac{55}{4} = 10$

$x = \frac{13}{4}$

$y = 2 + x$

$\frac{55}{4} = 10$

$x = \frac{13}{4}$

$y = 2 + x$

$\frac{55}{4} = 10$

$x = \frac{13}{4}$

$y = 2 + x$

Step 4.3

Replace all occurrences of $x$ in $y = 2 + x$ with $\frac{13}{4}$ .

$y = 2 + \frac{13}{4}$

$\frac{55}{4} = 10$

$x = \frac{13}{4}$

Step 4.4

Simplify $y = 2 + \frac{13}{4}$ .

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Step 4.4.1

Simplify the left side.

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Step 4.4.1.1

Remove parentheses.

$y = 2 + \frac{13}{4}$

$\frac{55}{4} = 10$

$x = \frac{13}{4}$

$y = 2 + \frac{13}{4}$

$\frac{55}{4} = 10$

$x = \frac{13}{4}$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Simplify $2 + \frac{13}{4}$ .

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Step 4.4.2.1.1

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = 2 \cdot \frac{4}{4} + \frac{13}{4}$

$\frac{55}{4} = 10$

$x = \frac{13}{4}$

Step 4.4.2.1.2

Combine $2$ and $\frac{4}{4}$ .

$y = \frac{2 \cdot 4}{4} + \frac{13}{4}$

$\frac{55}{4} = 10$

$x = \frac{13}{4}$

Step 4.4.2.1.3

Combine the numerators over the common denominator.

$y = \frac{2 \cdot 4 + 13}{4}$

$\frac{55}{4} = 10$

$x = \frac{13}{4}$

Step 4.4.2.1.4

Simplify the numerator.

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Step 4.4.2.1.4.1

Multiply $2$ by $4$ .

$y = \frac{8 + 13}{4}$

$\frac{55}{4} = 10$

$x = \frac{13}{4}$

Step 4.4.2.1.4.2

Add $8$ and $13$ .

$y = \frac{21}{4}$

$\frac{55}{4} = 10$

$x = \frac{13}{4}$

$y = \frac{21}{4}$

$\frac{55}{4} = 10$

$x = \frac{13}{4}$

$y = \frac{21}{4}$

$\frac{55}{4} = 10$

$x = \frac{13}{4}$

$y = \frac{21}{4}$

$\frac{55}{4} = 10$

$x = \frac{13}{4}$

$y = \frac{21}{4}$

$\frac{55}{4} = 10$

$x = \frac{13}{4}$

$y = \frac{21}{4}$

$\frac{55}{4} = 10$

$x = \frac{13}{4}$

Step 5

Since $\frac{55}{4} = 10$ is not true, there is no solution.

No solution

Step 6