Transcript Chapter 2
Chapter 2 Solving Linear Equations 2.1 Writing Equations 4 Steps to Problem Solving – – – – Explore the problem (read the whole thing) Plan the solution (write the equation) Solve the problem Check the solution (does it make sense?) How to use a Formula – – – – Write the formula Substitute for any known variables Solve the equation Check the answer (does it make sense?) Perimeter=2l+2w Area of a Square= lw Area of a Triangle= ½ bh Area of a Circle= r2 2.2 Solving Equations by Using Addition and Subtraction To solve an equation means to find all values of the variable that make it true Addition Property of Equality: – If you add the same amount to each side of the equal sign, the equation is true Ex: m – 48 = 29 Subtraction Property of Equality: – If you subtract the same amount from each side, the equation is true 42 + d = 27 n + 5 = 40 2.3 Solving Equations by Using Multiplication and Division Multiplication Property of Equality: – If you multiply each side of the equal sign by the same number, the equation is true t/3 = 7 9/4g = 1/2 Division Property of Equality: – If you divide each side of the equal sign by the same number, the equation is true 13s = 195 -3x = 36 2.4 Solving Multi-Step Equations Add or Subtract the number farthest from the variable Multiply or divide the number next to the variable Simplify Check your answer Examples: (p – 15)/9 = -6 2/3y – 25 = 115 Even consecutive and odd consecutive numbers = x, x+2, x+4, x+6…. Consecutive numbers = x, x+1, x+2, x+3… – – Find three consecutive integers whose sum is 21 Find three consecutive even integers whose sum is -42 2.5 Solving Equations with Variables on Each Side Distribute and/or combine like terms Add or subtract the variables to one side (move the smaller one) Add or subtract the numbers to the other side Solve as normal Examples: -2 + 10k = 8k -1 2m = 5 = 5(m – 7) -3m 2.6 Ratios and Proportions Ratio: a comparison of two numbers by division – x x to y x:y y ex: Your class has 21 students, 9 are boys and 12 are girls a. ratio of boys to girls ________________ b. ratio of students to boys _____________ Scale: a ratio that shows that a model is proportional to an actual object Proportion: shows that two ratios are equal – – a and d are the extremes b and c are the means a c b d Solve a proportion by cross multiplying – – a c . b d ad = bc Ex: Determine if it is a proportion. Ex: Solve the proportion. – a. n 24 15 16 – b. 2 w 12 6 9 15 35 36 42 2.7 Percent of Change New # is greater than original # = % of increase New # is less than original # = % of decrease To solve: new original #100 % original Original: $25 New: $28 Original: 16 New: 3 Concert tickets cost $45 each. The tax is 6.25%. What is the total cost for one ticket? A sweater is on sale for 35% off. The original price is $38. What is the sale price? 2.8 Solving for a Specific Variable Use the normal order of operations and problem solving steps to get the specific variable on one side of the equal sign and everything else on the other side of the equal sign Solve 3x – 4y = 7 for y. Solve 2m – t = sm + 5 for m. Solve C=2 r for r Solve s 1 2 at 2 for a. 2.9 Weighted Averages Weighted average: the sum of a product of units and value per unit, divided by the sum of the # of units How many pounds of mixed nuts selling for $4.75 per pound should be mixed with 10 pounds of dried fruit selling for $5.50 per pound, to obtain trail mix that sells for $4.95 per pound? # Units Dried Fruit 10 Price per Unit 5.50 Nuts x 4.75 4.75x Trail Mix 10 + x 4.95 4.95(10 + x) 550 55.0 + 4.75x = 4.95(10 + x) SOLVE FOR X Product An experiment calls for 30% solution of copper sulfate. Kendra has 40ml of 25% solution. How many ml of 60% solution should be added? 25% Solution 60% Solution 30% Solution Amount/Units 40 x 40 + x 10 + .6x = .3(40 + x) SOLVE FOR X Product 40 x .25= 10 .6x .3(40 + x)