Translate among common forms of a line equation

Ontario Grade 10 Mathematics

Analytic Geometry

Recognize what each form shows and rewrite it without changing the line

One line can be described by several equations. The equations may look different, but they can still match the same points on a graph. Each common form makes certain information easier to see. In this lesson, you will review slope and coordinates, learn what each form shows, and practise translating between forms while keeping the equation balanced.

What you will learn

Recognize slope-intercept, point-slope, and standard form.
Explain what information each form displays.
Translate an equation into another common form using balanced equation steps.
Check that two forms describe the same line.

1. Grade 9 bridge: points, slope, and balance

A point on a graph is written as an ordered pair, such as

(2,5)

. The first number is the

x

-coordinate and the second is the

y

-coordinate. The coordinates tell you the point's position on the graph.

Slope describes how a line rises or falls as you move from left to right. It compares the change in

y

with the change in

x

. A positive slope rises from left to right; a negative slope falls. Slope is often named with the letter

m

.

To find a slope from two points, divide the change in the

y

-coordinates by the change in the

x

-coordinates. The change is the difference between coordinate values. For example, from

(1,2)

to

(3,8)

, the changes are

6

in

y

and

2

in

x

, so the slope is

3

.

An equation is balanced when its left side and right side have equal values. Adding, subtracting, multiplying, or dividing both sides by the same non-zero number keeps the equation balanced. These steps help you rewrite an equation without changing its solutions.

m=\frac{\Delta y}{\Delta x}

An ordered pair lists the $x$ -coordinate first and the $y$ -coordinate second.
Slope describes a line's steepness and direction.
Use the same operation on both sides to keep an equation balanced.

2. Three common forms

Slope-intercept form is written as

y=mx+b

. The letter

m

represents the slope. The letter

b

represents the

y

-intercept, which is the

y

-value where the line crosses the vertical axis. For example, in

y=2x+3

, the slope is

2

and the

y

-intercept is

3

.

Point-slope form uses a known point and the slope. It is written as

y-y_1=m(x-x_1)

. The symbols

x_1

and

y_1

name the coordinates of the known point. The small subscript

1

is part of the name; it does not mean multiplication. For a point

(2,5)

, the matching coordinates are

x_1=2

and

y_1=5

.

Standard form is commonly written as

Ax+By=C

. The letters

A

,

B

, and

C

stand for numbers. This form groups the terms with variables on one side and a constant on the other. A question may give additional rules about signs or coefficients, so follow those instructions when they are provided.

A form is a way of arranging an equation. Slope-intercept form makes the slope and intercept easy to see. Point-slope form displays a slope and a point. Standard form groups the variable terms together.

y-y_1=m(x-x_1)

Slope-intercept form shows the slope and the $y$ -intercept.
Point-slope form shows a slope and a point on the line.
Standard form groups the variable terms on one side.

3. Translating between forms

Start by identifying the form you have and the form you want. Then choose steps that reveal the needed information. If you know a slope and a

y

-intercept, slope-intercept form is direct. If you know a point and a slope, point-slope form records them together.

To change point-slope form to slope-intercept form, distribute the slope across the brackets. Then use balanced equation steps to get

y

by itself. The result can be read as

y=mx+b

.

To write an equation in standard form, collect the variable terms on one side and put the constant on the other. If you reverse the sides of an equation or multiply both sides by

-1

, equality is preserved. Keep track of every sign.

A translated equation should describe the same relationship between

x

and

y

as the original. You can check this by substituting a point known to be on the line. If the equation is correct, its two sides have equal values for that point.

Ax+By=C

Expand brackets carefully, including negative signs.
Keep an equation balanced while rearranging it.
Check a translated equation with a known point.

4. Guided example

A line has slope

3

and passes through

(2,5)

. Since a slope and a point are given, begin with point-slope form. To make slope-intercept form, expand the brackets and isolate

y

. To make standard form, collect the variable terms and arrange the equation with a constant on the other side.

The detailed steps show why each change is valid. When you multiply both sides by

-1

, every term on both sides changes sign. This gives a standard-form equation for the same line.

Match the point's coordinates to the correct positions in the form.
Distribute before isolating $y$ .
When changing to standard form, show operations on both sides.

5. Independent practice

Try translating

y-4=-2(x-3)

into slope-intercept form, then write it in standard form. The equation gives a slope of

-2

and the point

(3,4)

. Expand the brackets, simplify, and rearrange the variable terms. Check your result by substituting

(3,4)

.

As you practise, name the starting form and the target form before doing the algebra. This helps you choose the right steps. Show enough work to track the signs and keep the equation balanced. After simplifying, use the point from the original equation as a check.

Use the given form to identify the information you already have.
Show steps that make signs and balance clear.
A translated equation should still work for a known point.

At a glance: common line forms

Form	Typical equation	Information shown
Slope-intercept	$y=mx+b$	Slope $m$ and $y$ -intercept $b$
Point-slope	$y-y_1=m(x-x_1)$	Slope $m$ and point $(x_1,y_1)$
Standard	$Ax+By=C$	Variable terms grouped on one side

Worked example

Translate a point-slope equation

A line has slope

3

and passes through

(2,5)

. Write its equation in point-slope, slope-intercept, and standard form.

Substitute the point and slope
Point-slope form uses the slope and the differences from the point's coordinates. Put $3$ in for $m$ , $2$ in for $x_1$ , and $5$ in for $y_1$ .
$y-5=3(x-2)$
Expand and isolate y
Distribute $3$ across the bracket. Then add $5$ to both sides so that $y$ is by itself. The result is slope-intercept form.
$y-5=3x-6\quad\Rightarrow\quad y=3x-1$
Rearrange into standard form
Subtract $3x$ from both sides to collect the variable terms. This gives $y-3x=-1$ . Now multiply both sides by $-1$ . Every term changes sign, giving an equivalent equation in standard form.
$y-3x=-1\quad\Rightarrow\quad 3x-y=1$

Answer: Point-slope form:

y-5=3(x-2)

. Slope-intercept form:

y=3x-1

. Standard form:

3x-y=1

.

Check: Substitute

(2,5)

into the slope-intercept equation:

3(2)-1=5

. Into standard form:

3(2)-5=1

. The point satisfies both equations.

Common mistakes and how to avoid them

Reading

b

as the slope in slope-intercept form.

Correction: In

y=mx+b

, the slope is

m

. The

y

-intercept is

b

.

Changing a sign when moving a term without keeping the equation balanced.

Correction: Add or subtract the same quantity on both sides, then simplify. If you multiply both sides by

-1

, change the sign of every term.

Putting the point's coordinates in the wrong places in point-slope form.

Correction: For

(x_1,y_1)

, use

x-x_1

inside the brackets and

y-y_1

on the left.

Assuming that equations that look different must describe different lines.

Correction: Simplify the equations or substitute a known point. Equivalent forms can look different but describe the same line.

Lesson summary

Slope-intercept form is $y=mx+b$ ; it displays slope and the $y$ -intercept.
Point-slope form is $y-y_1=m(x-x_1)$ ; it displays slope and a point.
Standard form commonly groups terms as $Ax+By=C$ .
Expand, collect terms, and keep both sides balanced when translating.
Check a translation by substituting a known point or simplifying the equations.

Check your understanding

Question 1

Which equation is in point-slope form and shows slope

-2

through

(4,1)

?

$y-1=-2(x-4)$
$y=-2x+1$
$2x+y=4$
$y+4=-2(x-1)$

Show answer and explanation

y-1=-2(x-4)

Substitute

m=-2

,

x_1=4

, and

y_1=1

into point-slope form to get

y-1=-2(x-4)

.

Question 2

Which equation is equivalent to

y=2x+3

in standard form?

$2x-y=-3$
$2x+y=3$
$x-2y=3$
$2x-y=3$

Show answer and explanation

2x-y=-3

Subtract

2x

from both sides to get

y-2x=3

. Multiply both sides by

-1

to get

2x-y=-3

.

Key terms

Slope: A number that describes a line's steepness and direction.
Y-intercept: The point where a line crosses the vertical axis; its $y$ -coordinate is $b$ in slope-intercept form.
Equivalent equations: Equations with the same solutions, even when written differently.
Standard form: A common way to write a line equation with its variable terms grouped, such as $Ax+By=C$ .

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