Solving Linear Equations Tutorial 3 d A Solution

Solving Linear Equations Tutorial 3 d

A Solution Set n Consider the different meanings of the word solution. n The solution to the mystery escaped him. n n The town’s solution to its landfill problem is to encourage recycling. n n The word solution here refers to an explanation. Solution here refers to a method of solving a problem. A chemist mixes two solutions to obtain a 15% acid solution. n Solution here refers to a homogeneous molecular mixture

Solution Set In Mathematics we also have different kinds of solutions and, therefore, different kinds of solution sets. n Study the table below: n Equation/Inequality 3 x + 5 = 14 |x| 5 x+3=x-7 (x + 6)(x – 3)=0 Solution Set {3} {-5 x 5} No Solution {-6, 3} These examples illustrate that a solution set may have one member, more than one member, or no members.

Solving: Addition & Subtraction Equations One way to solve an equation is to get the variable alone on one side of the equal sign. n You can do this by using inverse operations, which are operations that undo one another. n Addition and subtraction are inverse operations. n You can use subtraction to undo addition and addition to undo subtraction. n

Solving: Addition & Subtraction Equations Example #1: n n n Solve the equation x + 4 = 7 Think to yourself: What is being done to the variable (x)? A 4 is being added to the variable (x). Subtraction undoes addition therefore you should subtract a 4 on the left to get x alone on one side. However, whatever you do to one side of an equation you must also do to the other side. x+4=7 -4 -4 x =3 Always check your answers! x + 4 = 7; Does x = 3? 3 + 4 = 7 is true therefore x = 3!

Solving: Addition & Subtraction Equations Example #2: n n n Solve the equation x - 12 = 20 Think to yourself: What is being done to the variable (x)? A 12 is being subtracted from the variable (x). Addition undoes subtraction, therefore you should add a 12 on the left to get x alone on one side. However, whatever you do to one side of an equation you must also do to the other side. x - 12 = 20 +12 x = 32 Always check your answers! x - 12 = 20; Does x = 32? 32 - 12 = 20 is true therefore x = 32!

Problem Solving: A veterinary assistant holds a dog and steps on a scale. The scale reads 193. 7 lb. Alone, the assistant weighs 135 lb. To find the weight of the dog, solve the equation w + 135 = 193. 7 n Think to yourself: What is being done to the variable (w)? -135 n A 135 is being added to the variable (w). w = 58. 7 n Subtraction undoes addition therefore The dog weighs 58. 7 lb. you should subtract a 135 on the left to get w alone on one side. Always check your answers! However, whatever you do to one side of w + 135 = 193. 7; Does w = 58. 7? an equation you must also do to the 58. 7 + 135 = 193. 7 is true! other side.

Solving: Multiplication & Division Equations Multiplication and division are inverse operations. n You can use division to undo multiplication and multiplication to undo division. n

Solving: Multiplication & Division Equations Example #1: n n n Solve the equation 5 x = 35 Think to yourself: What is being done to the variable (x)? A 5 is being multiplied to the variable (x). Division undoes multiplication, therefore you should divide a 5 to the left side to get x alone on that side. However, whatever you do to one side of an equation you must also do to the other side. 15 x = 357 5 5 1 1 x =7 Always check your answers! 5 x = 35; Does x = 7? 5 • 7 = 35 is true therefore x = 7 !

Solving: Multiplication & Division Equations Example #2: n n n Solve the equation Think to yourself: What is being done to the variable (r)? A 6 is being divided into the variable (r). Multiplication undoes division, therefore you should multiply a 6 to the left side to get r alone on that side. However, whatever you do to one side of an equation you must also do to the other side. 1 6 • • 6 1 r = 24 Always check your answers!

Solving: Multiplication & Division Equations Example #2: n n n Solve the equation Think to yourself: What is being done to the variable (r)? A 5/6 is being multiplied to the variable (r). Multiplying by the reciprocal will eliminate the fraction, therefore you should multiply a 6/5 to the left side to get r alone on that side. However, whatever you do to one side of an equation you must also do to the other side. 1 1 18 r = 108 Always check your answers! 1