# Different Ways of Writing a Line

A line travels through the points $$(0,6)$$ and $$(2,2).$$ What is the equation of the line?

## Different Ways of Writing a Line

### Equations and Unknowns

# Different Ways of Writing a Line

Some of the graphing we have done has been with equations that are written in the form $$y=mx + b$$, where $$m$$ is the slope of the line and $$b$$ is the $$y$$-intercept; this is slope-intercept form. (The previous problem involved slope-intercept form. For more practice with slope-intercept form, visit the Equations of Lines quiz.)

However, we can write equations of lines in many different forms, and the slope-intercept form isn't always the most appropriate choice. While all the different forms of equations produce equivalent graphs, some forms allow you to see certain graph characteristics more clearly.

## Different Ways of Writing a Line

### Equations and Unknowns

# Different Ways of Writing a Line

One advantage of slope-intercept form is that it makes it easy to spot parallel lines.

Select all of the equations that produce parallel lines.

## Different Ways of Writing a Line

### Equations and Unknowns

Select one or more.

# Different Ways of Writing a Line

Both the volleyball and soccer teams at Fairview High School are trying to raise money. The volleyball team currently has $0 and is hoping to raise$100 per month. The soccer team has $1000 and is hoping to raise$50 per month.

The volleyball team's fundraising is out-pacing that of the soccer team. When the two teams have the same amount of money, how much money will they each have?

## Different Ways of Writing a Line

### Equations and Unknowns

# Different Ways of Writing a Line

The standard form of a linear equations is written as $Ax + By = C.$ If the slope-intercept form of a line is $$y = \frac{2}{3}x + 4,$$ what is the standard form?

Note that you can choose to use algebra to determine the solution, or work it out with the visualization below.

## Different Ways of Writing a Line

### Equations and Unknowns

# Different Ways of Writing a Line

If a line is represented by both $y = mx + b$ and $Ax+By=C,$ is it possible that $$m=A$$ and $$b=C?$$ $$($$Note that $$A, B,$$ and $$C$$ are non-zero integers.$$)$$

## Different Ways of Writing a Line

### Equations and Unknowns

# Different Ways of Writing a Line

Suppose that we rewrote $$Ax + By = C$$ by dividing both sides by $$C:$$ $\frac{A}{C}x + \frac{B}{C}y = 1 .$ What are the $$x$$-intercept and $$y$$-intercept of this line?

## Different Ways of Writing a Line

### Equations and Unknowns

# Different Ways of Writing a Line

The visualization below lets you play with $$h$$ and $$k$$ in the equation $y = m(x-h) + k.$ What is true about the point $$(h,k) ?$$

## Different Ways of Writing a Line

### Equations and Unknowns

# Different Ways of Writing a Line

The previous form, point-slope form, is often written as $$y = m(x-h) + k .$$ It is useful when we want to use the slope, $$m,$$ and the coordinates of a point, $$(h,k),$$ to write the equation of a line.

For example, if a line has a slope of $$\frac{2}{3}$$ and travels through the point $$(6,5),$$ we can write the equation of the line as $y=\frac{2}{3}(x-6)+5.$ If we want to continue simplifying into slope-intercept form, we get $y=\frac{2}{3}x-4+5 = \frac{2}{3}x+1.$ Note that sometimes point-slope form is written as $y - y_0 = m(x-x_0),$ but that the above form is more useful when working with functions.

## Different Ways of Writing a Line

### Equations and Unknowns

# Different Ways of Writing a Line

Which equation represents the line that travels through the points $$(10,60)$$ and $$(15,100)?$$

## Different Ways of Writing a Line

### Equations and Unknowns

Select one or more.

# Different Ways of Writing a Line

While there are many different forms of equations of lines, the three that we have covered are the most commonly used.

• Slope-intercept form is ideal when you know the slope and $$y$$-intercept of a line. $$m$$ represents the slope and $$b$$ represents the $$y$$-intercept: $y=mx+b.$
• Standard form is ideal when you want to easily convey the $$x$$- and $$y$$-intercepts of a line. $$A, B,$$ and $$C$$ must be integers and generally $$A\neq 0$$ and $$B \neq 0.$$ The $$x$$-intercept is $$\left(\frac{C}{A}, 0\right)$$ and the $$y$$-intercept is $$\left(0, \frac{C}{B}\right):$$ $Ax+By=C.$
• Point-slope form is ideal when you know the slope and any point on the line. $$m$$ is the slope of the line and $$(h,k)$$ are the coordinates of any point on the line: $y=m(x-h)+k.$

## Different Ways of Writing a Line

### Equations and Unknowns

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