Lesson Explainer: Equation of a Straight Line: General Form | Nagwa Lesson Explainer: Equation of a Straight Line: General Form | Nagwa

# Lesson Explainer: Equation of a Straight Line: General Form Mathematics • First Year of Secondary School

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In this explainer, we will learn how to find and write the equation of a straight line in general form.

We recall that the straight line with slope and -intercept is described by the equation

This is the slopeβintercept form of the equation of a straight line. There are many different ways to describe a line. For example, the equation of a line with slope passing through the point has the pointβslope form

All the different ways to represent a line as an equation have something in common. A point lies on the line if and only if the equation of the line is satisfied when and . In this explainer, we introduce the general form of a straight line.

### Definition: General Form of the Equation of a Straight Line

The general form of the equation of a straight line is given by where , , and are real constants.

We remark that all lines can be written in the general form, while some equations of straight lines cannot be written in the pointβslope or slopeβintercept forms. For example, the line has no equation in slopeβintercept form since the slope of this line is undefined. Nonetheless, the equation can be written in general form: .

The general form of the equation of a straight line does not directly show the slope of the line. To obtain the slope of a line from its general form, it is beneficial to use the equation in slopeβintercept form: where represents the slope of the line. We can convert an equation in its general form to one in the slopeβintercept form by solving for the variable .

Given the equation with , we subtract and from both sides to get

Then, dividing both sides of the equation by leads to the slopeβintercept form

So, a line represented by the equation in its general form , if , has slope and -intercept .

Let us consider an example where we derive the slope of a line from the given equation in general form.

### Example 1: Finding the Slope of a Line given Its Equation in General Form

A straight line has the equation . What is the slope of the line?

We are given the equation of a line in general form:

To obtain the slope of the line, we should convert the above equation into the slopeβintercept form where is the slope of the line and is the -intercept. From the slopeβintercept form, we can identify the slope, which is given by .

To convert the equation to the slopeβintercept form, we should make the subject of the equation. Let us achieve this in steps. First, we add and 12 to both sides of the equation to get which leads to

Next, we divide both sides by 3 to get

So, we obtain the slopeβintercept form . This tells us the slope of the line is 5.

The straight line with the equation has a slope of 5.

In the next example, we obtain the - and -intercepts of a line from its equation in general form.

### Example 2: Finding the π₯- and π¦-Intercepts of a Line

What are the -intercept and -intercept of the line ?

Let us first consider the -intercept of the line. We are given the equation of the line in general form:

We recall that the -intercept of a line has coordinates of the form . So, the -intercept of the line must satisfy the equation above with . Substituting into the equation, we get

We then solve for :

Next, let us consider the -intercept of the line. We recall that the -intercept has coordinates of the form , so the -intercept must satisfy the equation with . Plugging into the equation gives us

We then solve for :

The line given by has -intercept 4 and -intercept 6.

We note that the right-hand side of the equation of a line in general form is equal to zero. If we multiply both sides of the equation in general form by any nonzero constant, we would still end up with an equation in general form. For example, if a straight line is given in general form as , then we can multiply through by 2 to get

So, is another equation of the same line in general form. This property is useful when simplifying an equation with rational coefficients. Say that we have the equation of a straight line in general form:

We can multiply both sides by 4 to cancel the denominator, 4:

Then, the equation of the same line in general form is more simply written as .

The general form of the equation of a line is closely related to its standard form: where , , and are integers and is nonnegative. We can convert the standard form into general form by subtracting the constant from both sides of the equation.

Let us consider different examples where we derive the equation of a straight line in general form. In the next example, we find the general form of the equation of a line from its slope and -intercept.

### Example 3: Finding the Equation of a Line in General Form given Its Slope and π¦-Intercept

Write the equation of the line with slope and -intercept in the form .

We are given the slope and -intercept of the line. We recall that the slopeβintercept form for the equation of this line is

We need to convert this equation into the form , which is known as the general form of the equation of the line. We simplify the coefficients by multiplying both sides of the equation by 2 to get

We can subtract from both sides of the equation to get

Then, we can rearrange the equation into the form :

Hence, the equation of the line with slope and -intercept can be written in general form as .

When we are given two points and on the line, we recall that the slope of the line is given by

Then, using one of the points , we can write the equation of the line in pointβslope form:

We can use either of the points and to obtain the equation in pointβslope form. Although the resulting equations will look different depending on the point chosen, they are both equivalent representations of the line. Often, though not always, either choice will lead to the same expression when converted to general form.

In the next example, we derive the equation of a straight line in general form given the coordinates of two points lying on the line.

### Example 4: Finding the Equation of a Line through Two Points Giving the Answer in a Specified Form

Find the equation of the line that passes through the points and , giving your answer in the form .

We recall that the slope of the line passing through points and can be computed by

We are given that the line passes through the points and . Letting be and be , we get

So, the slope of the line is .

To write the equation of this line, we also need a point on the line. We are given two points to choose from, and either choice is valid. For this problem, we notice that one of the points, , is actually the -intercept. So, choosing would lead to the slopeβintercept form of the equation, while choosing would lead to a pointβslope form of the equation. We will present both choices, and we will finish both methods by converting the resulting equations to the form .

Method 1

We note that the point is on the -axis, so it is the -intercept of the line. Since we know the slope of the line , we can write the equation of the line in slopeβintercept form:

We need to convert this equation to the form . Multiplying both sides of the equation by 10 gives us

Subtracting from both sides of the equation, we get

Finally, rearranging the equation to the form gives us

In order to match one of the provided options, we multiply both sides of this equation by :

Method 2

Let us use the point and the slope obtained above to write the equation of the line in pointβslope form:

Multiplying both sides by 10 and distributing the parenthesis, we get

Subtracting and 30 from both sides of the equation gives us

We note that this equation matches one of the provided options.

So, the equation of our line is given in option B.

Given the -intercept and the -intercept of a line where and are nonzero, we can write the equation of the line in two-intercept form:

We note that the -intercept lies on the line since and , giving

Likewise, we can check for the -intercept by substituting and , which gives us

This form is quite convenient since we do not need to compute the slope of the line to obtain it, and it is easy to sketch since we can mark both intercepts and connect them with a straight line as pictured below.

Since both expressions on the left-hand side of the equation in the two-intercept form are quotients, we begin the conversion process to its general form by multiplying both sides of the equation by the common denominator.

In the next example, we will derive the equation in general form when two intercepts are provided.

### Example 5: Finding the Equation of a Straight Line

Determine the equation of the line that cuts the -axis at 4 and the -axis at 7.

Since the line cuts the -axis at 4 and the -axis at 7, the - and -intercepts of the line are 4 and 7, respectively. Let us write the equation in the general form: . We present two different ways of approaching this problem. In the first method, we use the given intercepts to write the equation of the line in two-intercept form. In the second method, we use the intercepts to find the slope of the line, which is used to write the equation of the line in slopeβintercept form.

Method 1

Since the - and -intercepts of the line are 4 and 7, we can write the equation of the line in two-intercept form:

Multiplying both sides of the equation by the common denominator 28 gives us

By subtracting 28 from both sides of the equation, we obtain the general form

Method 2

Since the - and -intercepts of the line are 4 and 7, we know that the line passes through the two points and . We recall that the slope of the line is given by

So, the slope of the line is . We also know that the -intercept is 7. Then, we can write the equation of the line in slopeβintercept form:

We can simplify this equation by multiplying both sides by 4. Then,

Rearranging the equation gives us

So, the equation of the line that cuts the -axis at 4 and the -axis at 7 is

### Key Points

• The general form of the equation of a straight line is where , , and are constants.
• All straight lines can be represented by an equation in general form.
• A line represented by an equation in its general form , if , has slope and -intercept . The equation of this line in its slopeβintercept form is
• A line represented by the - and -intercepts in the form can be represented in general form by multiplying the whole equation by the common denominator and rearranging the terms:

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