Algebra I

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

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

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.

                     

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

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.

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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.\()\)

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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

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

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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

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

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: \[y=m(x-h)+k.\]
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