System of Linear Equations ?

Question

In economics, the IS curve is a linear equation that represents all combinations of income Y and interest rates r that maintain an equilibrium in the market for goods in the economy. The LM curve is a linear equation that represents all combinations of income Y and interest rates r that maintain an equilibrium in the market for money in the economy. In an economy, suppose the equilibrium level of income (in millions of dollars) and interest rates satisfy the system of equations.:

0.06Y - 6000r = 120

0.06Y + 7000r = 900

Find the equilibrium level of income and interest rates.

How is this done? 

Answer 1

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There are many methods to solving a system of two linear equations with two unknowns.   Since the coefficients of "y" are the same I'll use elimination.

0.06y - 6000r = 120

0.06y + 7000r = 900

If we multiply one equation by -1 we can then add both equations together to "eliminate" the y term.  I'll multiply the first one:

-0.06y + 6000r = -120

0.06y + 7000r = 900

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13000r = 780

r = 780 / 13000

r = 0.06

Now that we know r we can solve for y:

0.06y - 6000r = 120

0.06y - 6000(0.06) = 120

0.06y - 360 = 120

0.06y = 480

y = 480 / 0.06

y = 8000

The equilibrium income would be 8 billion dollars (8000 millions) with a rate of 6%.

Answer 2

alg 1 -- solve the equations by eliminating a variable

  0.06Y - 6000r = 120

-[0.06Y + 7000r = 900]

     -1300r = -780

          r  =  0.06  <<  use this to solve for Y

  0.06Y - 6000*0.06 = 120

   0.06Y = 480

    Y = 8000