Algebra Examples

$\frac{1}{2} x - 3 y = 8$ , $y = 4 x + 1$

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

Simplify the left side.

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Step 1.2.1

Combine $\frac{1}{2}$ and $x$ .

$\frac{x}{2} - 3 y = 8$

Step 1.3

Subtract $\frac{x}{2}$ from both sides of the equation.

$- 3 y = 8 - \frac{x}{2}$

Step 1.4

Divide each term in $- 3 y = 8 - \frac{x}{2}$ by $- 3$ and simplify.

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Step 1.4.1

Divide each term in $- 3 y = 8 - \frac{x}{2}$ by $- 3$ .

$\frac{- 3 y}{- 3} = \frac{8}{- 3} + \frac{- \frac{x}{2}}{- 3}$

Step 1.4.2

Simplify the left side.

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Step 1.4.2.1

Cancel the common factor of $- 3$ .

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Step 1.4.2.1.1

Cancel the common factor.

$\frac{- 3 y}{- 3} = \frac{8}{- 3} + \frac{- \frac{x}{2}}{- 3}$

Step 1.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{8}{- 3} + \frac{- \frac{x}{2}}{- 3}$

Step 1.4.3

Simplify the right side.

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Step 1.4.3.1

Simplify each term.

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Step 1.4.3.1.1

Move the negative in front of the fraction.

$y = - \frac{8}{3} + \frac{- \frac{x}{2}}{- 3}$

Step 1.4.3.1.2

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{8}{3} - \frac{x}{2} \cdot \frac{1}{- 3}$

Step 1.4.3.1.3

Move the negative in front of the fraction.

$y = - \frac{8}{3} - \frac{x}{2} (- \frac{1}{3})$

Step 1.4.3.1.4

Multiply $- \frac{x}{2} (- \frac{1}{3})$ .

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Step 1.4.3.1.4.1

Multiply $- 1$ by $- 1$ .

$y = - \frac{8}{3} + 1 \frac{x}{2} \frac{1}{3}$

Step 1.4.3.1.4.2

Multiply $\frac{x}{2}$ by $1$ .

$y = - \frac{8}{3} + \frac{x}{2} \cdot \frac{1}{3}$

Step 1.4.3.1.4.3

Multiply $\frac{x}{2}$ by $\frac{1}{3}$ .

$y = - \frac{8}{3} + \frac{x}{2 \cdot 3}$

Step 1.4.3.1.4.4

Multiply $2$ by $3$ .

$y = - \frac{8}{3} + \frac{x}{6}$

Step 1.5

Write in $y = m x + b$ form.

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Step 1.5.1

Reorder $- \frac{8}{3}$ and $\frac{x}{6}$ .

$y = \frac{x}{6} - \frac{8}{3}$

Step 1.5.2

Reorder terms.

$y = \frac{1}{6} x - \frac{8}{3}$

Step 2

Using the slope-intercept form, the slope is $\frac{1}{6}$ .

$m_{1} = \frac{1}{6}$

Step 3

Use the slope-intercept form to find the slope.

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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

Using the slope-intercept form, the slope is $4$ .

$m_{2} = 4$

Step 4

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

$\frac{1}{2} x - 3 y = 8, y = 4 x + 1$

Step 5

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

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Step 5.1

Replace all occurrences of $y$ with $4 x + 1$ in each equation.

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Step 5.1.1

Replace all occurrences of $y$ in $\frac{1}{2} x - 3 y = 8$ with $4 x + 1$ .

$\frac{1}{2} x - 3 (4 x + 1) = 8$

$y = 4 x + 1$

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Simplify $\frac{1}{2} x - 3 (4 x + 1)$ .

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Step 5.1.2.1.1

Simplify each term.

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Step 5.1.2.1.1.1

Combine $\frac{1}{2}$ and $x$ .

$\frac{x}{2} - 3 (4 x + 1) = 8$

$y = 4 x + 1$

Step 5.1.2.1.1.2

Apply the distributive property.

$\frac{x}{2} - 3 (4 x) - 3 \cdot 1 = 8$

$y = 4 x + 1$

Step 5.1.2.1.1.3

Multiply $4$ by $- 3$ .

$\frac{x}{2} - 12 x - 3 \cdot 1 = 8$

$y = 4 x + 1$

Step 5.1.2.1.1.4

Multiply $- 3$ by $1$ .

$\frac{x}{2} - 12 x - 3 = 8$

$y = 4 x + 1$

$\frac{x}{2} - 12 x - 3 = 8$

$y = 4 x + 1$

Step 5.1.2.1.2

To write $- 12 x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x}{2} - 12 x \cdot \frac{2}{2} - 3 = 8$

$y = 4 x + 1$

Step 5.1.2.1.3

Simplify terms.

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Step 5.1.2.1.3.1

Combine $- 12 x$ and $\frac{2}{2}$ .

$\frac{x}{2} + \frac{- 12 x \cdot 2}{2} - 3 = 8$

$y = 4 x + 1$

Step 5.1.2.1.3.2

Combine the numerators over the common denominator.

$\frac{x - 12 x \cdot 2}{2} - 3 = 8$

$y = 4 x + 1$

$\frac{x - 12 x \cdot 2}{2} - 3 = 8$

$y = 4 x + 1$

Step 5.1.2.1.4

Simplify each term.

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Step 5.1.2.1.4.1

Simplify the numerator.

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Step 5.1.2.1.4.1.1

Factor $x$ out of $x - 12 x \cdot 2$ .

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Step 5.1.2.1.4.1.1.1

Raise $x$ to the power of $1$ .

$\frac{x - 12 x \cdot 2}{2} - 3 = 8$

$y = 4 x + 1$

Step 5.1.2.1.4.1.1.2

Factor $x$ out of $x^{1}$ .

$\frac{x \cdot 1 - 12 x \cdot 2}{2} - 3 = 8$

$y = 4 x + 1$

Step 5.1.2.1.4.1.1.3

Factor $x$ out of $- 12 x \cdot 2$ .

$\frac{x \cdot 1 + x (- 12 \cdot 2)}{2} - 3 = 8$

$y = 4 x + 1$

Step 5.1.2.1.4.1.1.4

Factor $x$ out of $x \cdot 1 + x (- 12 \cdot 2)$ .

$\frac{x (1 - 12 \cdot 2)}{2} - 3 = 8$

$y = 4 x + 1$

$\frac{x (1 - 12 \cdot 2)}{2} - 3 = 8$

$y = 4 x + 1$

Step 5.1.2.1.4.1.2

Multiply $- 12$ by $2$ .

$\frac{x (1 - 24)}{2} - 3 = 8$

$y = 4 x + 1$

Step 5.1.2.1.4.1.3

Subtract $24$ from $1$ .

$\frac{x \cdot - 23}{2} - 3 = 8$

$y = 4 x + 1$

$\frac{x \cdot - 23}{2} - 3 = 8$

$y = 4 x + 1$

Step 5.1.2.1.4.2

Move $- 23$ to the left of $x$ .

$\frac{- 23 \cdot x}{2} - 3 = 8$

$y = 4 x + 1$

Step 5.1.2.1.4.3

Move the negative in front of the fraction.

$- \frac{23 x}{2} - 3 = 8$

$y = 4 x + 1$

$- \frac{23 x}{2} - 3 = 8$

$y = 4 x + 1$

$- \frac{23 x}{2} - 3 = 8$

$y = 4 x + 1$

$- \frac{23 x}{2} - 3 = 8$

$y = 4 x + 1$

$- \frac{23 x}{2} - 3 = 8$

$y = 4 x + 1$

Step 5.2

Solve for $x$ in $- \frac{23 x}{2} - 3 = 8$ .

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Step 5.2.1

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

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Step 5.2.1.1

Add $3$ to both sides of the equation.

$- \frac{23 x}{2} = 8 + 3$

$y = 4 x + 1$

Step 5.2.1.2

Add $8$ and $3$ .

$- \frac{23 x}{2} = 11$

$y = 4 x + 1$

$- \frac{23 x}{2} = 11$

$y = 4 x + 1$

Step 5.2.2

Multiply both sides of the equation by $- \frac{2}{23}$ .

$- \frac{2}{23} \cdot (- \frac{23 x}{2}) = - \frac{2}{23} \cdot 11$

$y = 4 x + 1$

Step 5.2.3

Simplify both sides of the equation.

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Step 5.2.3.1

Simplify the left side.

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Step 5.2.3.1.1

Simplify $- \frac{2}{23} (- \frac{23 x}{2})$ .

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Step 5.2.3.1.1.1

Cancel the common factor of $2$ .

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Step 5.2.3.1.1.1.1

Move the leading negative in $- \frac{2}{23}$ into the numerator.

$\frac{- 2}{23} \cdot (- \frac{23 x}{2}) = - \frac{2}{23} \cdot 11$

$y = 4 x + 1$

Step 5.2.3.1.1.1.2

Move the leading negative in $- \frac{23 x}{2}$ into the numerator.

$\frac{- 2}{23} \cdot \frac{- 23 x}{2} = - \frac{2}{23} \cdot 11$

$y = 4 x + 1$

Step 5.2.3.1.1.1.3

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{23} \cdot \frac{- 23 x}{2} = - \frac{2}{23} \cdot 11$

$y = 4 x + 1$

Step 5.2.3.1.1.1.4

Cancel the common factor.

$\frac{2 \cdot - 1}{23} \cdot \frac{- 23 x}{2} = - \frac{2}{23} \cdot 11$

$y = 4 x + 1$

Step 5.2.3.1.1.1.5

Rewrite the expression.

$\frac{- 1}{23} \cdot (- 23 x) = - \frac{2}{23} \cdot 11$

$y = 4 x + 1$

$\frac{- 1}{23} \cdot (- 23 x) = - \frac{2}{23} \cdot 11$

$y = 4 x + 1$

Step 5.2.3.1.1.2

Cancel the common factor of $23$ .

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Step 5.2.3.1.1.2.1

Factor $23$ out of $- 23 x$ .

$\frac{- 1}{23} \cdot (23 (- x)) = - \frac{2}{23} \cdot 11$

$y = 4 x + 1$

Step 5.2.3.1.1.2.2

Cancel the common factor.

$\frac{- 1}{23} \cdot (23 (- x)) = - \frac{2}{23} \cdot 11$

$y = 4 x + 1$

Step 5.2.3.1.1.2.3

Rewrite the expression.

$x = - \frac{2}{23} \cdot 11$

$y = 4 x + 1$

$x = - \frac{2}{23} \cdot 11$

$y = 4 x + 1$

Step 5.2.3.1.1.3

Multiply.

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Step 5.2.3.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 x = - \frac{2}{23} \cdot 11$

$y = 4 x + 1$

Step 5.2.3.1.1.3.2

Multiply $x$ by $1$ .

$x = - \frac{2}{23} \cdot 11$

$y = 4 x + 1$

$x = - \frac{2}{23} \cdot 11$

$y = 4 x + 1$

$x = - \frac{2}{23} \cdot 11$

$y = 4 x + 1$

$x = - \frac{2}{23} \cdot 11$

$y = 4 x + 1$

Step 5.2.3.2

Simplify the right side.

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Step 5.2.3.2.1

Simplify $- \frac{2}{23} \cdot 11$ .

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Step 5.2.3.2.1.1

Multiply $- \frac{2}{23} \cdot 11$ .

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Step 5.2.3.2.1.1.1

Multiply $11$ by $- 1$ .

$x = - 11 (\frac{2}{23})$

$y = 4 x + 1$

Step 5.2.3.2.1.1.2

Combine $- 11$ and $\frac{2}{23}$ .

$x = \frac{- 11 \cdot 2}{23}$

$y = 4 x + 1$

Step 5.2.3.2.1.1.3

Multiply $- 11$ by $2$ .

$x = \frac{- 22}{23}$

$y = 4 x + 1$

$x = \frac{- 22}{23}$

$y = 4 x + 1$

Step 5.2.3.2.1.2

Move the negative in front of the fraction.

$x = - \frac{22}{23}$

$y = 4 x + 1$

$x = - \frac{22}{23}$

$y = 4 x + 1$

$x = - \frac{22}{23}$

$y = 4 x + 1$

$x = - \frac{22}{23}$

$y = 4 x + 1$

$x = - \frac{22}{23}$

$y = 4 x + 1$

Step 5.3

Replace all occurrences of $x$ with $- \frac{22}{23}$ in each equation.

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Step 5.3.1

Replace all occurrences of $x$ in $y = 4 x + 1$ with $- \frac{22}{23}$ .

$y = 4 (- \frac{22}{23}) + 1$

$x = - \frac{22}{23}$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Simplify $4 (- \frac{22}{23}) + 1$ .

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Step 5.3.2.1.1

Simplify each term.

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Step 5.3.2.1.1.1

Multiply $4 (- \frac{22}{23})$ .

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Step 5.3.2.1.1.1.1

Multiply $- 1$ by $4$ .

$y = - 4 (\frac{22}{23}) + 1$

$x = - \frac{22}{23}$

Step 5.3.2.1.1.1.2

Combine $- 4$ and $\frac{22}{23}$ .

$y = \frac{- 4 \cdot 22}{23} + 1$

$x = - \frac{22}{23}$

Step 5.3.2.1.1.1.3

Multiply $- 4$ by $22$ .

$y = \frac{- 88}{23} + 1$

$x = - \frac{22}{23}$

$y = \frac{- 88}{23} + 1$

$x = - \frac{22}{23}$

Step 5.3.2.1.1.2

Move the negative in front of the fraction.

$y = - \frac{88}{23} + 1$

$x = - \frac{22}{23}$

$y = - \frac{88}{23} + 1$

$x = - \frac{22}{23}$

Step 5.3.2.1.2

Simplify the expression.

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Step 5.3.2.1.2.1

Write $1$ as a fraction with a common denominator.

$y = - \frac{88}{23} + \frac{23}{23}$

$x = - \frac{22}{23}$

Step 5.3.2.1.2.2

Combine the numerators over the common denominator.

$y = \frac{- 88 + 23}{23}$

$x = - \frac{22}{23}$

Step 5.3.2.1.2.3

Add $- 88$ and $23$ .

$y = \frac{- 65}{23}$

$x = - \frac{22}{23}$

Step 5.3.2.1.2.4

Move the negative in front of the fraction.

$y = - \frac{65}{23}$

$x = - \frac{22}{23}$

$y = - \frac{65}{23}$

$x = - \frac{22}{23}$

$y = - \frac{65}{23}$

$x = - \frac{22}{23}$

$y = - \frac{65}{23}$

$x = - \frac{22}{23}$

$y = - \frac{65}{23}$

$x = - \frac{22}{23}$

Step 5.4

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

$(- \frac{22}{23}, - \frac{65}{23})$

Step 6

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

$m_{1} = \frac{1}{6}$

$m_{2} = 4$

$(- \frac{22}{23}, - \frac{65}{23})$

Step 7