Precalculus Examples

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

Use the slope-intercept form to find the slope.

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

$y = m x + b$

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

$m_{1} = 6$

Use the slope-intercept form to find the slope.

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

$y = m x + b$

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

$m_{2} = 1$

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

$y = 6 x$

$y = x + 3$

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

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Eliminate the equal sides of each equation and combine.

$6 x = x + 3$

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

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Subtract $x$ from both sides of the equation.

$6 x - x = 3$

$y = x + 3$

Subtract $x$ from $6 x$ .

$5 x = 3$

$y = x + 3$

$5 x = 3$

$y = x + 3$

Divide each term by $5$ and simplify.

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

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

$y = x + 3$

Reduce the expression by cancelling the common factors.

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Cancel the common factor.

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

$y = x + 3$

Divide $x$ by $1$ .

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

$y = x + 3$

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

$y = x + 3$

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

$y = x + 3$

Replace all occurrences of $x$ with the solution found by solving the last equation for $x$ . In this case, the value substituted is $\frac{3}{5}$ .

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

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

Simplify the right side.

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To write $\frac{3}{1}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

$y = \frac{3}{5} + \frac{3}{1} \cdot \frac{5}{5}$

Write each expression with a common denominator of $5$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

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

$y = \frac{3}{5} + \frac{3 \cdot 5}{1 \cdot 5}$

Multiply $5$ by $1$ .

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

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

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

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

Combine the numerators over the common denominator.

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

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

Simplify the numerator.

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Multiply $3$ by $5$ .

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

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

Add $3$ and $15$ .

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

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

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

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

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

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

Solve for $y$ in the second equation.

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

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

The solution to the system of equations can be represented as a point.

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

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

$m_{1} = 6$

$m_{2} = 1$

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

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