EQUATIONS WITH LITERALCOEFFICIENTS
As stated earlier, the first letters of thealphabet usually represent known quantities (constants), and the last letters represent unknown quantities (variables). Thus, we usually solve for x, y , or 2.
An equation such as
ax - 8 = bx - 5
has letters as coefficients. Equations with literal coefficients are solved in the same way asequations with numerical coefficients, except that when an operation cannot actually be performed, it merely is indicated.
In solving for x in the equation
ax - 8 =bx-5
subtract bx from both members and add 8 toboth members. The result is
ax - bx = 8 - 5
Since the subtraction on the left side cannotactually be performed, it is indicated. The quantity, a - b, is the coefficient of x when terms are collected. The equation takes the form
(a - b) x = 3
Now divide both sides of the equation by a-b.Again the result can be only indicated. The solution of the equation is
In solving for y in the equation
ay + b = 4
subtract b from both members as follows:
ay = 4 - b
Dividing both members by a, the solution is
Practice problems. Solve for x in each ofthe following:
REMOVING SIGNS OF GROUPING
If signs of grouping appear in an equation they should be removed in the manner indicated in chapter 9 of this course. For example, solvethe equation.
5 = 24 - [x-12(x-2) - 6(x-2)]
Notice that the same expression, x-2, occurs inboth parentheses. By combining the terms containing (x-2), the equation becomes
5 = 24 - [x-18(x-2)]
Next, remove the parentheses and then thebracket, obtaining
Subtracting 17x from both members and thensubtracting 5 from both members, we have
Divide both members by -17. The solution is
EQUATIONS CONTAINING FRACTIONS
To solve for x in an equation such as
first clear the equation of fraction, To do this, find the least common denominator of the fraction. Then multiply both rider of the equation by the LCD. The least common denominator of 3, 12, 4, and 2 is 12. Multiply both rider of the equation by 12. The resulting equation is
8x + x - 12 - 3 + 8x
Subtract 6x from both members, add 12 to bothmembers, and collect like terms as follows:
The solution is
To prove that x = 5 is the correct solution,substitute 5 for x in the original equation and show that both sides of the equation reduce to the same value. The result of substitution is
In establishing an identity, the two sides ofthe equality are treated separately, and the op the equality, and it is desirable to find the least common denominator for more than one set of fractions. The same denominator could be used on both sides of the equality, but this might make some of the terms of the fractions larger than necessary.
Proceeding in establishing the identity forx = 5 in the foregoing equation we obtain
Each member of the equality has the value11/4 when x = 5. The fact that the equation be proves that x = 5 is the solution.
Practice problems. Solve each of the following equations: