Standard Form of a Line

Assumption	$x -$ intercept	$y -$ intercept
$A \neq = 0,$ $B \neq = 0$	$(\frac{C}{A}, 0)$	$(0, \frac{C}{B})$
$A = 0,$ $B \neq = 0$	The line is horizontal, $y = \frac{C}{B},$ so it does not cross the $x -$ axis.	$(0, \frac{C}{B})$
$A \neq = 0,$ $B = 0$	$(\frac{C}{A}, 0)$	The line is vertical, $x = \frac{C}{A},$ so it does not cross the $y -$ axis.

Assumption

x -

intercept

y -

intercept

A \neq = 0,

B \neq = 0

(\frac{C}{A}, 0)

(0, \frac{C}{B})

A = 0,

B \neq = 0

The line is horizontal,

y = \frac{C}{B},

so it does not cross the

x -

axis.

(0, \frac{C}{B})

A \neq = 0,

B = 0

(\frac{C}{A}, 0)

The line is vertical,

x = \frac{C}{A},

so it does not cross the

y -

axis.

a The graph of the relationship between the price of a school bag and the price of the pencil case is given.

The $y -$ intercept and the slope of the line is used to write the equation of the line in slope-intercept form. Let's first look at the $y -$ intercept. Remember that the $y -$ intercept of a line is the $y -$ coordinate of the point where the line crosses the $y -$ axis.

The $y -$ intercept of the line is $5 .$ Now find the ratio of the change in $y -$ values to the change in $x -$ values to find the slope.

When the

x -

values increase by

2,

y -

values increase by

5 .

Slope Slope = \frac{changes in y - value}{changes in x - value} ⇓ = \frac{5}{2}

The value of the slope is the coefficient of variable

x,

and the

y -

intercept is the constant of the equation.

Substitute the values into the equation to write the equation of the graph in slope-intercept form.

y y = m x + n ⇓ = \frac{5}{2} x + 5

b All terms for the variables

x

and

y

are on one side of the linear equation and the constant is on the other side in standard form.

A x + B y = C

Additionally, the coefficients of the variables do not equal

0

in the standard form of an equation and, the coefficient of

x

is greater than

0 .

Rearrange the terms in the equation written in Part A to write it in standard form.

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

MultEqn

$LHS \cdot 2 = RHS \cdot 2$

2 y = 2 \cdot \frac{5}{2} x + 10

▼

Multiply

MoveLeftFacToNum

$a \cdot \frac{b}{c} = \frac{a \cdot b}{c}$

2 y = \frac{2 \cdot 5}{2} x + 10

CancelCommonFac

Cancel out common factors

2 y = \frac{2 \cdot 5}{2} x + 10

SimpQuot

Simplify quotient

2 y = 5 x + 10

SubEqn

$LHS - 2 y + 10 = RHS - 2 y + 10$

2 y - (2 y + 10) = 5 x + 10 - (2 y + 10)

▼

Simplify equation

Distr

Distribute $- 1$

2 y - 2 y - 10 = 5 x + 10 - 2 y - 10

CommutativePropAdd

Commutative Property of Addition

2 y - 2 y - 10 = 5 x + 10 - 10 - 2 y

SubTerms

Subtract terms

- 10 = 5 x - 2 y

RearrangeEqn

Rearrange equation

5 x - 2 y = - 10

Note that the coefficients of the variables are

5

and

- 2

which are not equal to

0,

and the coefficient of

x

is greater than

0 .

Here is the equation in standard form.

5 x - 2 y = - 10

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Catch-Up and Review

Comparing the Slope-Intercept Form and Standard Form

Recognizing the Standard Form of a Line

Graphing a Linear Equation in Standard Form

Extra

Stationery Shopping

Answer

Hint

Solution

Writing a Linear Equation in Standard Form

Preparing for School

Hint

Solution

Comparing Prices

Hint

Solution

Finding the Values of $A,$ $B,$ and $C$

Converting the Slope-Intercept Form of a Line to Standard Form

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