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Video: Finding the Equation of a Line in General Form

Lauren McNaughten

Write the equation of the line that passes through the points (2, −2) and (−2, 10) in the form 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0.

03:30

Video Transcript

Write the equation of line that passes through the points two, negative two and negative two, 10 in the form 𝑎𝑥 plus 𝑏𝑦 plus 𝑐 is equal to zero.

So in this question, we’re asked to find the equation of a straight line given the coordinates of two points that lie on the line. We’ll answer this question using the slope-intercept method. First, we’ll find the equation of the straight line in the form 𝑦 equals 𝑚𝑥 plus 𝑐, where 𝑚 represents the slope of the line and 𝑐 represents the 𝑦-intercept. This won’t give the equation of the line in the specific form that’s been asked for: 𝑎𝑥 plus 𝑏𝑦 plus 𝑐 is equal to zero. So we’ll need to do some rearrangement at the end.

Let’s recall first how to calculate the slope of a line. The slope of the line joining the points with coordinates 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two is equal to the change in 𝑦 divided by the change in 𝑥: 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one.

Now, let’s substitute the coordinates in this question into the formula for the slope. 𝑦 two minus 𝑦 one is equal to 10 minus negative two. 𝑥 two minus 𝑥 one is equal to negative two minus two. This simplifies to 12 over negative four which is equal to negative three. So the slope of the line is negative three.

Substituting this value into our equation for the line gives 𝑦 is equal to negative three 𝑥 plus 𝑐. Next, we need to find the value of 𝑐 — the 𝑦-intercept. To do this, we can use the fact that either of these two points lie on the straight line and, therefore, their coordinates satisfy the equation of the line. Using the first point, this means that when 𝑥 is equal to two, 𝑦 is equal to negative two. And so we can substitute this pair of values into the equation of the line to give an equation that we can solve for 𝑐.

Substituting two for 𝑥 and negative two for 𝑦 gives the equation negative two is equal to negative three multiplied by two plus 𝑐. Now, we solve for 𝑐. Negative three multiplied by two is negative six. So we have negative two is equal to negative six plus 𝑐. To solve this equation, we need to add six to both sides. This gives four is equal to 𝑐.

Now, we know the value of 𝑐, we can substitute this into the equation of our line. We have 𝑦 is equal to negative three 𝑥 plus four. Remember we’re asked for the equation of a straight line in the form 𝑎𝑥 plus 𝑏𝑦 plus 𝑐 is equal to zero, so we need all of the terms on the same side. We could group them on either side. But if I grouped the terms on the left, then the coefficients of both 𝑥 and 𝑦 will be positive.

In order to group the terms on the left-hand side of this equation, I need to add three 𝑥 to both sides and also subtract four from both sides. This gives the equation of the straight line in the requested format: three 𝑥 plus 𝑦 minus four is equal to zero.