Geometry, Analytic Geometry, Line equations

[missIdle]

Please help me out with this problem : A straight line passes through the points (3,3) and (-2,-4). Write it's equation in intercept form.

I know how to solve it's equation given only one point. But when it comes to two points I'm having trouble getting the equation. The answer of this problem is x/4/3 - y/8/5 =1

Thank you in advance!

 

[HallsofIvy]

Please help me out with this problem : A straight line passes through the points (3,3) and (-2,-4). Write it's equation in intercept form.

I know how to solve it's equation given only one point. But when it comes to two points I'm having trouble getting the equation. The answer of this problem is x/4/3 - y/8/5 =1

This is ambiguous. Do you mean x/(4/3)- y/(8/5)= 1 or (x/4)/3- (y/8)/5= 1. The first would be better written 3x/4- 5y/8= 1 while the second would be x/12- y/40= 1.
If x= 3, the first is 9/4- 14/8= 18/8- 14/8= 4/8= 1/2, not 3. The second would be 3/12- 3/40= 1/4- 3/40= 10/4- 3/40= 7/40, still not correct.

NO, the answer is NOT 'x/4/3- y/8/5= 1 no matter what that means!

Thank you in advance! [/QUOTE]
It is impossible to do "given only one point" because there are an infinite number of different lines through one point. Did you mean "given one point and the slope? If so then surely you also know how to find slope, don't you? The slope of a line through (3, 3) and (-2, -4) is $\displaystyle \frac{3- (-4)}{3- (-2)}= \frac{7}{5}$. The line through (3, 3) with slope 7/5 is y- 3= (7/5)(x- 3). Similarly, the line through (-2, -4) with slope 7/5 is y- (-4)= (7/5)(x- (-2)).

From y- 3= (7/5)(x- 3), multiplying both sides by 5, 5y- 15= 7x- 21 or 5y- 7x= .-6
From y-(-4)= (7/5)(x- (-2)), which is the same as y+ 4= (7/5)(x+ 2), multiplying both sides by 5, 5y+ 20= 7x+ 14 or 5y- 7x= -6 the same as before.

Personally, I like to use the fact that any (non-vertical) line can be written "y= ax+ b" for some numbers a and b. If x= 3, y= 3 so that becomes 3= 3a+ b and if x= -2, y= -4 so that become -4= -2x+ b. Subtracting the second equation from the first eliminates "b": 3- (-4)= 3a+ b- (-2a+ b) or 7= 5a. a= 7/5 Then 3= 3(7/5)+ b so b= 3- 21/5= 15/5- 21/5= -6/5.
y= (7/5)x- 6/5 or, multiplying by 5, 5y= 7x- 6. That can be written 7x- 5y= 6.

 

[missIdle]

Thank you for your answer. I really appreciate it. My teacher told us that it should be in intercept form and that is x/a -y/b = 1 I have already tried the solution like what you did and ended up with the same equation, but I am really confused of that intercept form. I know the intercept form but how could I end up with the answer x/(4/3)-y/(8/5) =1 ?

 

[ksdhart2]

You can transform the equation of a line into the form x/a - y/b = 1 if that's what your class instructions tell you to do... but you'll never get that equation, no matter how hard you try because it's the wrong answer. If you, for some reason, don't believe Halls's algebraic explanation of why it's wrong, you can try graphing it yourself. You'll see that the line does not pass through the point (3,3). Whoever told you that was the answer was mistaken. Simple as that.

If you want to find what the answer actually is, remember that in the form x/a + y/b = 1, the value of a is the x-intercept, and b is the y-intercept. What does that suggest must happen to b if you want to instead use the form x/a - y/b = 1?

 

[lookagain]

My teacher told us that it should be in intercept form and that is x/a - y/b = 1

The intercept form of a line is x/a + y/b = 1.

The y-intercept for this problem is negative, so the equation can have a subtraction sign
in it as it turns out.