Precalculus Examples

$y = 6 x$ , $y = x + 3$

Step 1

Use the slope-intercept form to find the slope.

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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

Using the slope-intercept form, the slope is $6$ .

$m_{1} = 6$

Step 2

Use the slope-intercept form to find the slope.

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Step 2.1

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 2.2

Using the slope-intercept form, the slope is $1$ .

$m_{2} = 1$

Step 3

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

$y = 6 x, y = x + 3$

Step 4

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

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Step 4.1

Eliminate the equal sides of each equation and combine.

$6 x = x + 3$

Step 4.2

Solve $6 x = x + 3$ for $x$ .

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Step 4.2.1

Move all terms containing $x$ to the left side of the equation.

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Step 4.2.1.1

Subtract $x$ from both sides of the equation.

$6 x - x = 3$

Step 4.2.1.2

Subtract $x$ from $6 x$ .

$5 x = 3$

Step 4.2.2

Divide each term in $5 x = 3$ by $5$ and simplify.

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Step 4.2.2.1

Divide each term in $5 x = 3$ by $5$ .

$\frac{5 x}{5} = \frac{3}{5}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $5$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{3}{5}$

Step 4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{5}$

Step 4.3

Evaluate $y$ when $x = \frac{3}{5}$ .

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Step 4.3.1

Substitute $\frac{3}{5}$ for $x$ .

$y = (\frac{3}{5}) + 3$

Step 4.3.2

Substitute $\frac{3}{5}$ for $x$ in $y = (\frac{3}{5}) + 3$ and solve for $y$ .

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Step 4.3.2.1

Remove parentheses.

$y = \frac{3}{5} + 3$

Step 4.3.2.2

Remove parentheses.

$y = (\frac{3}{5}) + 3$

Step 4.3.2.3

Simplify $(\frac{3}{5}) + 3$ .

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Step 4.3.2.3.1

To write $3$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \frac{3}{5} + 3 \cdot \frac{5}{5}$

Step 4.3.2.3.2

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

$y = \frac{3}{5} + \frac{3 \cdot 5}{5}$

Step 4.3.2.3.3

Combine the numerators over the common denominator.

$y = \frac{3 + 3 \cdot 5}{5}$

Step 4.3.2.3.4

Simplify the numerator.

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Step 4.3.2.3.4.1

Multiply $3$ by $5$ .

$y = \frac{3 + 15}{5}$

Step 4.3.2.3.4.2

Add $3$ and $15$ .

$y = \frac{18}{5}$

Step 4.4

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

$(\frac{3}{5}, \frac{18}{5})$

Step 5

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

$m_{1} = 6$

$m_{2} = 1$

$(\frac{3}{5}, \frac{18}{5})$

Step 6

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