Finite Math Examples

$2 x - y = 3$ , $3 x + y = 7$

Step 1

Rewrite in slope-intercept form.

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Step 1.1

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Step 1.2

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

$- y = 3 - 2 x$

Step 1.3

Divide each term in $- y = 3 - 2 x$ by $- 1$ and simplify.

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Step 1.3.1

Divide each term in $- y = 3 - 2 x$ by $- 1$ .

$\frac{- y}{- 1} = \frac{3}{- 1} + \frac{- 2 x}{- 1}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{3}{- 1} + \frac{- 2 x}{- 1}$

Step 1.3.2.2

Divide $y$ by $1$ .

$y = \frac{3}{- 1} + \frac{- 2 x}{- 1}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Divide $3$ by $- 1$ .

$y = - 3 + \frac{- 2 x}{- 1}$

Step 1.3.3.1.2

Move the negative one from the denominator of $\frac{- 2 x}{- 1}$ .

$y = - 3 - 1 \cdot (- 2 x)$

Step 1.3.3.1.3

Rewrite $- 1 \cdot (- 2 x)$ as $- (- 2 x)$ .

$y = - 3 - (- 2 x)$

Step 1.3.3.1.4

Multiply $- 2$ by $- 1$ .

$y = - 3 + 2 x$

Step 1.4

Reorder $- 3$ and $2 x$ .

$y = 2 x - 3$

Step 2

Using the slope-intercept form, the slope is $2$ .

$m_{1} = 2$

Step 3

Rewrite in slope-intercept form.

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Step 3.1

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Step 3.2

Subtract $3 x$ from both sides of the equation.

$y = 7 - 3 x$

Step 3.3

Reorder $7$ and $- 3 x$ .

$y = - 3 x + 7$

Step 4

Using the slope-intercept form, the slope is $- 3$ .

$m_{2} = - 3$

Step 5

Set up the system of equations to find any points of intersection.

$2 x - y = 3, 3 x + y = 7$

Step 6

Solve the system of equations to find the point of intersection.

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Step 6.1

Subtract $3 x$ from both sides of the equation.

$y = 7 - 3 x$

$2 x - y = 3$

Step 6.2

Replace all occurrences of $y$ with $7 - 3 x$ in each equation.

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Step 6.2.1

Replace all occurrences of $y$ in $2 x - y = 3$ with $7 - 3 x$ .

$2 x - (7 - 3 x) = 3$

$y = 7 - 3 x$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Simplify $2 x - (7 - 3 x)$ .

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Step 6.2.2.1.1

Simplify each term.

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Step 6.2.2.1.1.1

Apply the distributive property.

$2 x - 1 \cdot 7 - (- 3 x) = 3$

$y = 7 - 3 x$

Step 6.2.2.1.1.2

Multiply $- 1$ by $7$ .

$2 x - 7 - (- 3 x) = 3$

$y = 7 - 3 x$

Step 6.2.2.1.1.3

Multiply $- 3$ by $- 1$ .

$2 x - 7 + 3 x = 3$

$y = 7 - 3 x$

$2 x - 7 + 3 x = 3$

$y = 7 - 3 x$

Step 6.2.2.1.2

Add $2 x$ and $3 x$ .

$5 x - 7 = 3$

$y = 7 - 3 x$

$5 x - 7 = 3$

$y = 7 - 3 x$

$5 x - 7 = 3$

$y = 7 - 3 x$

$5 x - 7 = 3$

$y = 7 - 3 x$

Step 6.3

Solve for $x$ in $5 x - 7 = 3$ .

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Step 6.3.1

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

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Step 6.3.1.1

Add $7$ to both sides of the equation.

$5 x = 3 + 7$

$y = 7 - 3 x$

Step 6.3.1.2

Add $3$ and $7$ .

$5 x = 10$

$y = 7 - 3 x$

$5 x = 10$

$y = 7 - 3 x$

Step 6.3.2

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

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Step 6.3.2.1

Divide each term in $5 x = 10$ by $5$ .

$\frac{5 x}{5} = \frac{10}{5}$

$y = 7 - 3 x$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Cancel the common factor of $5$ .

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Step 6.3.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{10}{5}$

$y = 7 - 3 x$

Step 6.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{10}{5}$

$y = 7 - 3 x$

$x = \frac{10}{5}$

$y = 7 - 3 x$

$x = \frac{10}{5}$

$y = 7 - 3 x$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

Divide $10$ by $5$ .

$x = 2$

$y = 7 - 3 x$

$x = 2$

$y = 7 - 3 x$

$x = 2$

$y = 7 - 3 x$

$x = 2$

$y = 7 - 3 x$

Step 6.4

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

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Step 6.4.1

Replace all occurrences of $x$ in $y = 7 - 3 x$ with $2$ .

$y = 7 - 3 \cdot 2$

$x = 2$

Step 6.4.2

Simplify the right side.

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Step 6.4.2.1

Simplify $7 - 3 \cdot 2$ .

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Step 6.4.2.1.1

Multiply $- 3$ by $2$ .

$y = 7 - 6$

$x = 2$

Step 6.4.2.1.2

Subtract $6$ from $7$ .

$y = 1$

$x = 2$

$y = 1$

$x = 2$

$y = 1$

$x = 2$

$y = 1$

$x = 2$

Step 6.5

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

$(2, 1)$

Step 7

Since the slopes are different, the lines will have exactly one intersection point.

$m_{1} = 2$

$m_{2} = - 3$

$(2, 1)$

Step 8