Algebra Know-It-ALL

650 Worked-Out Solutions to Exercises: Chapters 11 to 19

Now let’s switch the left and right sides of the equation to get

mx=y−b

We can divide through by m, provided m≠ 0, and then use the right-hand distributive

law of division over subtraction to obtain

x=y/m−b/m

If we want it in strict SI form, we can rewrite it as

x= (1/m)y−b/m

Because x=f−^1 (y), we have

f−^1 (y)= (1/m)y−b/m

The slope of the inverse function is 1/m, and the x-intercept is −b/m. These values have

meaning only when m≠ 0. But if m≠ 0, f−^1 is always a function. A straight line with

defined slope (that is, a nonvertical line) in Cartesian coordinates can never produce more

than one value of the dependent variable for any single value of the independent variable.

Draw some sample graphs, and you’ll see why this is true. If you’re really ambitious, you

might try to formally prove it!

Chapter 18

1. Stated again for reference, the first and third revised equations are

− 4 x+ 2 y− 3 z= 5

and

3 x+ 6 y− 7 z= 0

We can multiply the top equation through by 7 to get

− 28 x+ 14 y− 21 z= 35

We can multiply the bottom equation through by −3 to get

− 9 x− 18 y+ 21 z= 0

When we add these two new equations in their entirety, we obtain the sum

− 28 x+ 14 y− 21 z= 35

− 9 x− 18 y+ 21 z= 0

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

− 37 x− 4 y= 35