Unit 4 - Equations Section 2 Solving One-Step Equations In this section we will be looking for solutions to equations. A solution is a number that can be plugged into an equation that keeps the equation true. Is =3 a solution to +7=10? +7=10 3+7=10 10=10 Since this is a true statement, = is a solution to + =. Is =2 a solution to +7=10? +7=10 2+7=10 9=10 Since 9 10, = is NOT a solution of + =. We can solve equations and find solutions as long as we keep the equations balanced as we work the problem. Think of the = a the tip of a balance or a scale. Adding pebbles to one side of a scale would through it off balance. In order to not change the balance you have to add pebbles to the other side of the scale as well. American River College 165 Milano
When I solve equations the first thing I do is look for my variable and think about how the numbers are attached. I then do the opposite to unattach them. Solving Equations of the Form + = For problems that have a number added to the variable I do the opposite of add I subtract the number from both sides. Example 1) Solve + = + 4 = 15 4 4 +0= 11 =11 We can Subtract 4 to get x by itself, but we must do it to both sides of the = sign. The Solution is 11. We can check to be sure this is correct by plugging =11 into the original problem. +4=15 11+4=15 15=15 Since the result is the same on both sides the solution is correct. For problems that have a number subtracted from the variable I do the opposite of subtract I add the number to both sides. Example 2) Solve = 7 = 12 +7 +7 + 0 = 19 We can add 7 to get y by itself, but we must do it to both sides of the = sign. =19 Check: This solution is correct since 19 7 = 12. American River College 166 Milano
Example 3) = 3 = 15 +3 +3 +0 = 12 We can add 3 to get b by itself, but we must do it to both sides of the = sign. = 12 Check: This solution is correct since 12 3= 15. Solving Equations of the Form = For problems that have a number multiplied to the variable I do the opposite of multiply I divide by the number on both sides. Example 4) = 5 =25 5 5 1 =5 We can divide by 5 to get x by itself, but we must do it to both sides of the = sign. =5 Check: The solution is correct since 5 5=25. Example 5 ) = 7 =42 7 7 1 = 6 We can divide by -7 to get y by itself, but we must do it to both sides of the = sign. = 6 Check: The solution is correct since 7 6=42. American River College 167 Milano
Example 6) = 3 = 36 3 3 We can divide by 5 to get a by itself, but we must do it to both sides of the = sign. =12 Check: The solution is correct since 3 12= 36. What do you do to get x by itself? Example 7) Consider the following problems a) = b) = a) = Since the 2 is held to the x by multiply I will do the opposite of multiply and divide. 2 = 10 2 = 10 2 2 1 = 5 x= 5 Check: 2 5=10 ) = Since the 10 is being subtracted from the x I will do the opposite of subtract and add. x 10= 15 x 10= 15 +10 +10 x + 0 = 5 Check: 5 10= 15 x= 5 American River College 168 Milano
If there are like terms on the same side of the =, we begin by combining them before considering what operation to do to both sides. Example 8) + = +8=12 5 We combine like terms before getting x by itself. +8 =7 8 8 +0= 1 = 1 Check: The solution is correct since 1+8=12 5. Example 9) = 7 3 =70 46 First combine like terms. 4 =24 4 4 =6 Check: The solution is correct since 7 6 3 6=70 46. Words to Equations It is essential to understand how words can build equations. Consider the following: A number decreased by 7 is 15. 7=15 American River College 169 Milano
Words that mean math add to the list as you come across new words! + - Add to Increased by Decreased by X Product of Quotient Divided Among Twice (X 2) Half ( 2) = is Example10) The product of a number and three is twenty seven, find the number. Equation: 3 =27 Product means multiply 3 and the number n. 3 =27 3 3 =9 The number is 9. Check: 3 9=27 American River College 170 Milano
Exercise 4.2 NAME: Solve the following. 1. +4=7 2. +7=17 3. +8=22 4. 13=10 5. 4=15 6. 3=7 7. +10= 15 8. 3= 10 9. 1= 12 American River College 171 Milano
10. 2 =12 11. 3 =24 12. 5 =45 13. 4 =16 14. 7 =42 15. 3 = 15 16. 4 = 24 17. 9 = 72 18. 8 =32 American River College 172 Milano
Solve 19. 3= 13 20. 4 = 16 21. 2 = 18 22. 5 3 =10 23. 4 7 = 33 24. 5 +8 =27 25. 5 = 72+47 26. 9 = 42 3 27. 3 =25 16 28. 10 +12 =13 7 29. 4 +3 5 =54 12 American River College 173 Milano
30. 15 7 = 12 52 Write an equation for the following and then solve. 31. Twice a number is 16, find the number. Equation: 32. A number increased by 20 is 35, find the number. Equation: 33. The product of a number and 5 is 55, find the number. Equation: American River College 174 Milano