Precalculus Examples

$y = x - 4$ , $y = 5 x$

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 $1$ .

$m_{1} = 1$

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 $5$ .

$m_{2} = 5$

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

$y = x - 4$

$y = 5 x$

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

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Eliminate the equal sides of each equation and combine.

$x - 4 = 5 x$

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

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Subtract $5 x$ from both sides of the equation.

$x - 4 - 5 x = 0$

$y = 5 x$

Subtract $5 x$ from $x$ .

$- 4 x - 4 = 0$

$y = 5 x$

$- 4 x - 4 = 0$

$y = 5 x$

Add $4$ to both sides of the equation.

$- 4 x = 4$

$y = 5 x$

Divide each term by $- 4$ and simplify.

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

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

$y = 5 x$

Reduce the expression by cancelling the common factors.

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Cancel the common factor.

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

$y = 5 x$

Divide $x$ by $1$ .

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

$y = 5 x$

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

$y = 5 x$

Divide $4$ by $- 4$ .

$x = - 1$

$y = 5 x$

$x = - 1$

$y = 5 x$

Solve for $x$ in the first equation.

$x = - 1 y = 5 x$

Replace all occurrences of $x$ with the solution found by solving the last equation for $x$ . In this case, the value substituted is $- 1$ .

$x = - 1$

$y = 5 (- 1)$

Multiply $5$ by $- 1$ .

$x = - 1$

$y = - 5$

Solve for $y$ in the second equation.

$x = - 1 y = - 5$

List the solutions to the system of equations.

$x = - 1$

$y = - 5$

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

$(- 1, - 5)$

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

$m_{1} = 1$

$m_{2} = 5$

$(- 1, - 5)$

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