SOLVING EQUATIONS WITH VARIABLES ON BOTH SIDES 1. Simplify each side Get rid of parentheses, add like terms, etc. 2. Get rid of variable on right side Add or subtract or use symmetric property. 3. Solve two step equation Add or subtract; then multiply or divide.

☺ SOLVING EQUATIONS WITH VARIABLES ON BOTH SIDES EXAMPLE GET RID OF THE VARIABLE ON THE RIGHT SIDE 5x – 3 = 3x + 7 -3x -3x 2x – 3 = 7 SOLVE 2-STEP EQUATION +3 +3 2x = 10 check: 5(5) – 3 = 3(5) + 7 2 2 25 – 3 = 15 + 7 22 = 22 x = 5 ☺

☺ SOLVING EQUATIONS WITH VARIABLES ON BOTH SIDES EXAMPLE 7 = 7(x – 3) SIMPLIFY BOTH SIDES 7 = 7(x – 3) GET RID OF THE VARIABLE ON THE RIGHT SIDE 7 = 7x – 21 7x – 21 = 7 SYMMETRIC PROPERTY SOLVE 2-STEP EQUATION +21 +21 7x = 28 check: 7 = 7(4 – 3) 7 7 7 = 7(1) 7 = 7 x = 4 ☺

☺ SOLVING EQUATIONS WITH VARIABLES ON BOTH SIDES EXAMPLE SIMPLIFY BOTH SIDES 2(m – 3) + 5 = 3(m – 1) 2m – 1 = 3m - 3 GET RID OF THE VARIABLE ON THE RIGHT SIDE -3m -3m -m – 1 = -3 SOLVE 2-STEP EQUATION +1 +1 check -m = -2 2(2 – 3) + 5 = 3(2 – 1) -1 -1 -2 + 5 = 3 ☺ m = 2

♠ ♣ ♠ ♣ ♠ ♣ ♠ ♠ A RATIO is a comparison of two numbers by division. RATIOS & PROPORTIONS A RATIO is a comparison of two numbers by division. Using the diagram at right, what is the ratio of spades to clubs? ♠ ♣ ♠ ♣ ♠ ♣ 5:3 or 5/3 or 5 to 3 ♠ ♠

A RATIO is a comparison of two numbers by division. RATIOS & PROPORTIONS A RATIO is a comparison of two numbers by division. In a recipe, if you are to use 2 cups of flour and 1 teaspoon of baking soda. What is the ratio of flour to baking soda? 2 cups:1 tsp Units must be included

A RATIO is a comparison of two numbers by division. RATIOS & PROPORTIONS A RATIO is a comparison of two numbers by division. If Warren can read 120 pages in two days, what is the ratio of pages read to days. 60 pages:1 day The ratio of two measurements having different measures is called a RATE.

Other examples of rates. RATIOS & PROPORTIONS Other examples of rates. 200 miles in 4 hours 50 miles/hour $5.20 for 24 eggs $2.60 / dozen $48 for 15 gallons of gas $3.20 / gallon

A PROPORTION is an equation stating that two ratios are equal. RATIOS & PROPORTIONS A PROPORTION is an equation stating that two ratios are equal. means extremes In a proportion, the product of the means is equal to the product of the extremes.

How do we know it is a proportion? RATIOS & PROPORTIONS How do we know it is a proportion? Cross Multiply (Means = Extremes) 20 20 Yes, it is a proportion!

How do we know it is a proportion? RATIOS & PROPORTIONS How do we know it is a proportion? Cross Multiply (Means = Extremes) 40 39 No, it is not a proportion!

Cross Multiply (Means = Extremes) RATIOS & PROPORTIONS Solving a proportion Cross Multiply (Means = Extremes) 21 2x 2x = 21 x = 21/2

Cross Multiply (Means = Extremes) RATIOS & PROPORTIONS Solving a proportion Cross Multiply (Means = Extremes) 15 2x - 4 2x - 4 = 15 2x = 19 x = 19/2

PROBLEM SOLVING USING PROPORTIONS If Jenny can travel 520 miles in 9 hours, how far can she travel in 15 hours? x = distance traveled in 15 hours 7800 9x 9x = 7800 x = 866.7 miles 866.7 miles

PROBLEM SOLVING USING PROPORTIONS If Al can read 1240 words in 3 minutes, how many words can he read in 20 minutes? x = words read in 20 minutes 3x 24800 3x = 24800 x = 8266.7 words 8266.7 words

PRACTICE – Variables on both sides Solve: 3(x + 2) = 4(x – 5) + 2 3x + 6 = 4x – 20 + 2 3x + 6 = 4x – 18 -4x -4x -x + 6 = -18 -6 -6 -x = -24 x = 24

PRACTICE – Ratios & Proportions Solve: Cross Multiply 4x + 20 = 3x - 6 -3x -3x x + 20 = -6 -20 -20 x = -26