Ch.10 SOLVING LINEAR EQUATIONS

TYPE II. Linear equations with the unknown in the numerator and/or denominator.

A typical form of Type II equations is as follows: ax + b e ----- = ---- cx + d fx+g where a, c and constant terms are not zero. The beginning strategy for this and other variations of the equation is to remove or eliminate the denominators by multiplyin both sides of the equation by the LCD and simplify. The resulting equation will conform to one of the forms of Type I linear equations. Thus, the remainder of the solution procedure follows what is indicated for Type I equations. However, a critical final step in the procedure must be observed. It is necessary to check the final constant value of the variable by substituting it in the original equation to determine if it really is the solution

TYPE III: Linear equations with the unknown under a radical sign

The property of squaring both sides of an equation is used as the first step in the solution procedure of this type of linear equation. When the result is simplified, this step should have eliminated all radical signs from the variable in the equation. However, if all of the radical signs are not removed from the variable, then usually it is necessary to rewrite the equation by transposing appropriately and again squaring both sides of the equation and simplify. Once the radical signs are removed, then the remainder of the solution procedure follows what is given above Type I equations. It is necessary to check the final result in the original equation to determine if it is a solution.

TYPE 1: Linear equations with the unknown in the numerator

This category of linear equations consists of a number of general forms and variations. Three of these forms are the equations ax + b = c, ax + b = dx + c, and a(x+e)=f

10-4

To solve an equation of form ax + b = dx + c where a and d are not zero, the beginning strategy is to rewrite the equation so that only one variable terms exists in the equation. This is done by adding the opposite of the variable term on the right side to both sides of the equation and simplifying. With only one variable term, the next step is to add the opposite of the constant term on the left hand side of the equation to both sides and simplify so that only one constant term remains. Finally, multiply each side of the equation by the reciprocal of the coefficient of the variable and simplify. The result is the equation variable = constant which is the solution

10-1

When solving a linear equation, the object is to simplify the equation using one or more of the four fundamental operations such that the equation has its final form variable = constant. The constant is the solution

10-5

When solving an equation involving one or more sets of parentheses a(x+e) =f where a and e are not zero, the first step in the solution procedure is to apply the distributive property to remove the parentheses. The remainder of the solution procedure is as indicated for the equation of the form ax +b = c

10-3

When solving an equation of form ax + b = c, where a does not = 0 the first step is to add the opposite of the constant b to each side of the equation and simplify. If coefficient a =1, then the final equation is in the form variable = constant which is the solution. On the other hand, if coefficient a does not = 1 and a does not = 0 then the next step is to remove the coefficient from the variable by multiplying each side of the equation by the coefficient and simplify. If coefficient a and other constant terms in the equation involve fractions, then it may be easier to multiply both sides of the equation by the LCD and simplify. In any case, the result is the form variable = constant which is the solution.