+2446 To convert from millimeters to inches in 2 dimensions, you must convert both the width and length before multiplying. I happen to only know this conversion from memorization:
\(1in=0.0254m\)
Of course, we want to go from millimeters to inches, so I must change the meters to millimeters. That is relatively simple:
| \(\frac{1000mm}{1m}=\frac{xmm}{0.0254m}\) | Using this proportion, we can figure out how many millimeters are in 0.0254 meters. |
| \(x=0.0254*1000=25.4mm\) | |
Ok, so now we know information that is a tad more useful than before.
\(1in=25.4mm\)
Using this knowledge, we can now convert both dimensions: 25mm and 40mm.
| \(\frac{1in}{25.4mm}=\frac{xin}{25mm}\) | Cross multiply to solve this proportion for x. |
| \(25=25.4x\) | Divide by 25.4 on both sides. |
| \(x=\frac{25}{25.4}*\frac{10}{10}\) | I, personally, do not like seeing decimals in fractions, so I am manipulating the fraction such that one exist. |
| \(x=\frac{250}{254}=\frac{125}{127}in\) | For now, I will leave the fraction in this form/ |
Now I will convert 40mm into inches:
| \(\frac{1in}{25.4mm}=\frac{yin}{40mm}\) | Cross multiply to solve this proportion for y. I changed the variable so that it would not be confusing. |
| \(40=25.4y\) | Divide by 25.4 on both sides. |
| \(y=\frac{40}{25.4}*\frac{10}{10}\) | Yet again, I am getting decimals out of the fraction. |
| \(y=\frac{400}{254}=\frac{200}{127}in\) | |
Ok, now multiply both of these fractions together.
\(\frac{125}{127}*\frac{200}{127}=\frac{25000}{16129}in^2\)
Unfortunately, this fraction is irreducible
\(\frac{25000}{16129}in^2\approx1.5500in^2\)
.+2446 To determine the intercepts of this equation in the form \(ax+by+c=0\), I would just make x and y zero and see what the result is:
Finding the x-intercept
For a line without the slope of 0, the line only touches the x-axis once. For a point to be on the x-intercept, y must be zero; otherwise, it would not be on the x-intercept. Knowing this, you can set the y to be zero and solve for x:
| \(4x+5y+6=0\) | Make y=0 so that the point is on the x-intercept. |
| \(4x+5*0+6=0\) | |
| \(4x+6=0\) | Subtract 6 on both sides. |
| \(4x=-6\) | Divide by 4 on both sides. |
| \(x=\frac{-6}{4}=-\frac{3}{2}=-1.5\) | |
Ok, we have determined that the x-intercept is located exactly on the point \((-\frac{3}{2},0)\)
Finding the y-intercept
You will utilize the exact same logic to find the y-intercept. Of course, x will be equal to 0 this time:
| \(4x+5y+6=0\) | Substitute 0 in for x. |
| \(4*0+5y+6=0\) | |
| \(5y+6=0\) | Subtract 6 on both sides. |
| \(5y=-6\) | Divide by 5 on both sides. |
| \(y=-\frac{6}{5}=-1.2\) | |
Ok, we have determined that the y-intercept is located exactly on the point \((0,-1.2)\).
You actually do not need any more information to graph this equation. Plot both the intercepts on a coordinate plane, and use a ruler to connect them. Then, you are done!
+2446 To figure out the equation of a line that passes through the given points (-6, -5) and (-4, -4), you must first know the standard form of a line. It is the following:
\(y=mx+b\)
Let m = slope of the line
Let b = the y-intercept (the point where the line touches the y-axis)
The first step is to figure out the slope of the line. How do we do that, you may ask? All you do is remember the slope formula.
\(m=\frac{y_2-y_1}{x_2-x_1}\)
We already have enough information to calculate the slope, m. We do this by substituting the given points into the formula.
| \(m=\frac{-5-(-4)}{-6-(-4)}\) | Simplify the fraction into simplest terms by evaluating the numerator and denominator separately. |
| \(m=\frac{-5+4}{-6+4}\) | Of course, subtracting a negative is the same as adding a positive. |
| \(m=\frac{-1}{-2}\) | The negatives in the numerator and denominator cancel each other out. |
| \(m=\frac{1}{2}\) | |
Great! We know the slope! Now, the only variable to figure out next is b, the y-intercept. We can do this by plugging in points of points on the line in the equation.
\(y=\frac{1}{2}x+b\)
In other words, to solve for b, you must plug in a point we know is one the line (either (-6,-5) or (-4,-4)) for x and y. I'll choose (-4,-4):
| \(y=\frac{1}{2}x+b\) | Plug in the coordinate (-4,-4) in its appropriate spots and then solve for b. |
| \(-4=\frac{1}{2}*-4+b\) | Now, solve for b. |
| \(-4=-2+b\) | Add 2 on both sides. |
| \(-2=b\) | |
Now that we have solved for both m and b, the equation that passes through the points (-6,-5) and (-4,-4) is \(y=\frac{1}{2}x-2\).
Do you need your answer in point-slope form? No problem! Remember the point-slope form
\(y-y_1=m(x-x_1)\)
Of course, m is the slope again. We have already calculated that. Let's substitute that in.
\(y-y_1=\frac{1}{2}(x-x_1)\)
\(y_1\hspace{1mm}\text{and}\hspace{1mm}x_1\) represent a point on the line. You can either substitute the first or the second set of coordinates. It doesn't matter. However, in the end, your answer should be one of these:
\(y+5=\frac{1}{2}(x+6)\)
\(y+4=\frac{1}{2}(x+4)\)
