Linear And Exponential Algebra Lesson #1

Transcription

1 Introduction Linear And Eponential Algebra Lesson # Algebra is a very powerful tool which is used to make problem solving easier. Algebra involves using pronumerals (letters) to represent unknown values or values which can vary depending on the situation. Pronumerals are also called variables. Many worded problems can be converted to algebraic symbols to make algebraic equations. We learn how to solve equations in order to find the solutions to problems. Algebra can also be used to construct formulae, which are equations that connect two or more variables. Many people use formulae as part of their jobs, so an understanding of how to substitute into and rearrange formulae is essential. Builders, nurses, pharmacists, engineers, financial planners and computer programmers all use formulae which rely on algebra. Algebraic Notation The ability to convert worded sentences and problems into algebraic symbols and to understand algebraic notation is essential in the problem solving process. Notice that: o + is an algebraic epression, whereas o + 8 is an equation, and o + > 8 is an inequality (sometimes called an inequation). Recall that when we simplify repeated sums, we use product notation: o i.e., + and + + lots of lots of Also, when we simplify repeated products, we use inde notation: o i.e., and

2 Algebraic Substitution Consider the number crunching machine to the right: If we place any number, into the machine, it calculates 7, i.e., is multiplied by and then 7 is subtracted. For eample: o if, 7 if -, 7 7 (-) To evaluate an algebraic epression, we find its value for particular numerical substitutions of the unknowns. Eample : If p 4, q - and r, find the value of: p q+ r (a.) q r (b.) pq r (c.) p+ r (a.) q r (b.) pq r (-) 4 (-) - -9 (c.) p q+ r p+ r 4 ( ) + 4+ ( ) Eample : If a, b - and c -, evaluate: (a.) b (b.) ab c (a.) b (b.) ab c (-) (-) (-) 4 -

3 Eample : If p 4, q - and r, calculate: (a.) p q+ r (b.) p+ q (a.) p q+ r (b.) Linear Equations p+ q 4 ( ) + 4+ ( ).6 { significant figures} Many problems can be written as equations using algebraic notation. So, it is essential we are able to solve equations. Linear equations are equations which are in the form (or can be converted to the form) a + b 0, where is the unknown (variable) and a, b are constants. Solving Equations The following steps should be followed when solving simple equations: o Step : Decide how the epression containing the unknown has been built up. o Step : Isolate the unknown by performing inverse operations on both sides of the equation to undo the build up in reverse order. o Step : Check your solution by substitution. Eample : Solve for : (a.) 4 7 (b.) 6 (a.) {adding to both sides} {dividing both sides by 4} 4 4 Check: 4 9 7

4 (b.) 6 6 {subtracting from both sides} {dividing both sides by -} Check: Eample : Solve for : (a.) + 6 (b.) ( ) (a.) + + {adding to both sides} {multiplying both sides by } 0 Check: 0 ( ) ( ) {multiplying both sides by } + + {adding to both sides} (b.) ( )

5 Check: ( ) ( ) Sometimes the unknown will appear more than once. When this occurs, we epand brackets, collect like terms, and then solve the equation. Eample : Solve for : ( ) ( ) ( ) ( ) 6 + {epanding brackets} 4 {collecting like terms} {adding to both sides} {dividing both sides by 4} Check: ( 4 ) ( 4 ) If the unknown appears on both sides of the equation, we: o epand any brackets and collect like terms o move the unknown to one side of the equation and the remaining terms to the other side o simplify and solve the equation. Eample 4: Solve for : (a.) (b.) 4 ( + ) (a.) {subtracting from both sides} {adding 4 to both sides} Check: LH S 0 4 6, RHS

6 (b.) 4 ( + ) 4 6 {epanding} + + {adding to both sides} 4 4 {dividing both sides by 4} 4 4 Check: LHS RHS Sometimes when more complicated equations are epanded, a linear equation results. Eample : 4+ + Solve for : ( ) ( )( ) ( ) ( 4+ )( + ) {epanding each side} {subtracting from both sides} {adding 6 to both sides} {subtracting 8 from both sides} {dividing both sides by }

7 Fractional Equations Fractional equations can be simplified by finding the least common denominator (LCD) of the fractions. Each term is then multiplied by the fraction which makes the denominators the same (LCD) and then the numerators are equated. Consider the following fractional equations: LCD is 6 LCD is ( ) 0 7 Eample : Solve for : + LCD is ( ) ( ) + has LCD 0 + {to create a common denominator} ( + ) {equating numerators} 6 + {epanding brackets} 6 + {taking from both sides} 6 {dividing both sides by } Eample : Solve for : has LCD {to create a common denominator} 6 {equating numerators}

8 6 Eample : Solve for : has LCD 4( ) {to create a common denominator} 4( + ) ( ) {equating numerators } {epanding brackets} {adding to both sides} {subtracting 4 from both sides} {dividing both sides by } Using Technology You could use a graphing package or a graphics calculator to solve linear equations. For eample: to solve we: o Enter the functions Y. 4. and Y 7.4 into the calculator. o Adjust the viewing window settings for values between -0 and 0 and y values between -0 and 0. o Graph the two straight lines. o Find the point of intersection of the two lines using the built in function. So,.6.