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# Working with linear equations

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This is the first question of a 4 part problem. I just need help with how to start it.
above is the 1st question to below problem:

In some experiments, tathered data suggest that volume of gas and temperature are related quantities. In one particular experiment, at a temp. of -30 degrees C, the volume of a gas was measured to be 380 cm3, while at a temp of 30 degrees C this same gas had a volume of 470 cm3. Assumer that the relationship between volume and temp. is linear. (question for this is Give a linear equation that expresses volume,V, in terms of Temp, T.)
We were told to do this without a calculator and w/a calculator.
Please help! I just don't know how to do the first question. If i get this one maybe I can figure out the others.
Rachel

https://brainmass.com/math/linear-algebra/working-linear-equations-8031

## SOLUTION This solution is FREE courtesy of BrainMass!

We have to fit these values into a straight line of the form

y = mx + c
where m and c are constants to be determined.

(We will take y = V and x = T )

(x1,y1)=(T1,V1) =(-30,380) and

(x2,y2)= (T2,V2) = (30,470)

We can do this by least squares fitting. To fit a given set of data in to a straight line of the form y = mx + c, the normal equations for the constants m and c are, (we will determine m and c from the following equations)

Sum(y) = n*c + m * Sum(x) ......(1)

and Sum(xy) = c * Sum(x) + m * Sum(x^2) ......(2)

where n is the number of data points (x1,y1), (x2,y2) etc.
here n = 2

Sum(y) is the sum of all the y values, here Sum(y) = 850

Sum(x) = (-30)+ (30) = 0

Sum(xy) = -30*380 + 30*470 = 2700

Sum(x^2) = (-30)^2 + (30)^2 = 1800

Using these equations in (1) and (2), we get two equations

(1) gives, 850 = 2 c

(2) gives 2700 = 1800 * m

from this, c = 425 and m = 1.5

Therefore the required linear equation is y = 1.5 * x + 425

Or, V = 1.5 * T + 425

You can verify this equation by putting the values of T in this and obtain the corresponding V value. Remember that this procedure is valid only if the relation between variables is linear.

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