Algebra Examples

2 x + 3 y = 5

$2 x + 3 y = 5$

Step 1

Solve for

y

$y$ .

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Step 1.1

Subtract

2 x

$2 x$ from both sides of the equation.

3 y = 5 - 2 x

$3 y = 5 - 2 x$

Step 1.2

Divide each term in

3 y = 5 - 2 x

$3 y = 5 - 2 x$ by

3

$3$ and simplify.

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Step 1.2.1

Divide each term in

3 y = 5 - 2 x

$3 y = 5 - 2 x$ by

3

$3$ .

\frac{3 y}{3} = \frac{5}{3} + \frac{- 2 x}{3}

$\frac{3 y}{3} = \frac{5}{3} + \frac{- 2 x}{3}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of

3

$3$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{5}{3} + \frac{- 2 x}{3}$

Step 1.2.2.1.2

Divide

$y$ by

$1$ .

$y = \frac{5}{3} + \frac{- 2 x}{3}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

$y = \frac{5}{3} - \frac{2 x}{3}$

Step 2

Rewrite in slope-intercept form.

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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

Reorder

$\frac{5}{3}$ and

$- \frac{2 x}{3}$ .

$y = - \frac{2 x}{3} + \frac{5}{3}$

Step 2.3

Write in

$y = m x + b$ form.

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Step 2.3.1

Reorder terms.

$y = - (\frac{2}{3} x) + \frac{5}{3}$

Step 2.3.2

Remove parentheses.

$y = - \frac{2}{3} x + \frac{5}{3}$

Step 3

Use the slope-intercept form to find the slope and y-intercept.

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Step 3.1

Find the values of

$m$ and

$b$ using the form

$y = m x + b$ .

$m = - \frac{2}{3}$

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

Step 3.2

The slope of the line is the value of

$m$ , and the y-intercept is the value of

$b$ .

Slope:

$- \frac{2}{3}$

y-intercept:

$(0, \frac{5}{3})$

Slope:

$- \frac{2}{3}$

y-intercept:

$(0, \frac{5}{3})$

Step 4

Any line can be graphed using two points. Select two

$x$ values, and plug them into the equation to find the corresponding

$y$ values.

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Step 4.1

Write in

$y = m x + b$ form.

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Step 4.1.1

Reorder

$\frac{5}{3}$ and

$- \frac{2 x}{3}$ .

$y = - \frac{2 x}{3} + \frac{5}{3}$

Step 4.1.2

Reorder terms.

$y = - (\frac{2}{3} x) + \frac{5}{3}$

Step 4.1.3

Remove parentheses.

$y = - \frac{2}{3} x + \frac{5}{3}$

Step 4.2

Find the x-intercept.

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Step 4.2.1

To find the x-intercept(s), substitute in

$0$ for

$y$ and solve for

$x$ .

$0 = - \frac{2}{3} x + \frac{5}{3}$

Step 4.2.2

Solve the equation.

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Step 4.2.2.1

Rewrite the equation as

$- \frac{2}{3} x + \frac{5}{3} = 0$ .

$- \frac{2}{3} x + \frac{5}{3} = 0$

Step 4.2.2.2

Simplify each term.

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Step 4.2.2.2.1

Combine

$x$ and

$\frac{2}{3}$ .

$- \frac{x \cdot 2}{3} + \frac{5}{3} = 0$

Step 4.2.2.2.2

Move

$2$ to the left of

$x$ .

$- \frac{2 x}{3} + \frac{5}{3} = 0$

Step 4.2.2.3

Subtract

$\frac{5}{3}$ from both sides of the equation.

$- \frac{2 x}{3} = - \frac{5}{3}$

Step 4.2.2.4

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$- 2 x = - 5$

Step 4.2.2.5

Divide each term in

$- 2 x = - 5$ by

$- 2$ and simplify.

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Step 4.2.2.5.1

Divide each term in

$- 2 x = - 5$ by

$- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 5}{- 2}$

Step 4.2.2.5.2

Simplify the left side.

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Step 4.2.2.5.2.1

Cancel the common factor of

$- 2$ .

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Step 4.2.2.5.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- 5}{- 2}$

Step 4.2.2.5.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{- 5}{- 2}$

Step 4.2.2.5.3

Simplify the right side.

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Step 4.2.2.5.3.1

Dividing two negative values results in a positive value.

$x = \frac{5}{2}$

Step 4.2.3

x-intercept(s) in point form.

x-intercept(s):

$(\frac{5}{2}, 0)$

x-intercept(s):

$(\frac{5}{2}, 0)$

Step 4.3

Find the y-intercept.

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Step 4.3.1

To find the y-intercept(s), substitute in

$0$ for

$x$ and solve for

$y$ .

$y = - \frac{2}{3} \cdot 0 + \frac{5}{3}$

Step 4.3.2

Solve the equation.

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Step 4.3.2.1

Remove parentheses.

$y = - \frac{2}{3} \cdot 0 + \frac{5}{3}$

Step 4.3.2.2

Simplify

$- \frac{2}{3} \cdot 0 + \frac{5}{3}$ .

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Step 4.3.2.2.1

Multiply

$- \frac{2}{3} \cdot 0$ .

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Step 4.3.2.2.1.1

Multiply

$0$ by

$- 1$ .

$y = 0 (\frac{2}{3}) + \frac{5}{3}$

Step 4.3.2.2.1.2

Multiply

$0$ by

$\frac{2}{3}$ .

$y = 0 + \frac{5}{3}$

Step 4.3.2.2.2

Add

$0$ and

$\frac{5}{3}$ .

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

Step 4.3.3

y-intercept(s) in point form.

y-intercept(s):

$(0, \frac{5}{3})$

y-intercept(s):

$(0, \frac{5}{3})$

Step 4.4

Create a table of the

$x$ and

$y$ values.

$\begin{array}{c} x & y \\ 0 & \frac{5}{3} \\ \frac{5}{2} & 0 \end{array}$

Step 5

Graph the line using the slope and the y-intercept, or the points.

Slope:

$- \frac{2}{3}$

y-intercept:

$(0, \frac{5}{3})$

$\begin{array}{c} x & y \\ 0 & \frac{5}{3} \\ \frac{5}{2} & 0 \end{array}$

Step 6