Download Presentation
## Solving Equations

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**Solving Equations**Solving open sentences using inverse operations.**What will happen if you add or subtract an equal amount of**weight on both sides of the scales?Solving equations is like balancing scales, we must always keep the sides equal.**Solving equations is just a matter of undoing operations**that are being done to the variable.In a simple equation, this may mean that we only have to undo one operation, as in the following example.Solve the following equation for xx + 3 = 8 x + 3 = 8 the variable is x x + 3 – 3 = 8 – 3we are adding 3 to the variable, so to get rid ofthe added 3, we do the opposite--- subtract 3. x = 5 remember to do this to both sides of theequation.**In an equation which has more than one operation, we have to**undo the operations in the correct order. We start with the operation the farthest away from the variable. Solve the following equation: 5x – 2 =13 5x – 2 = 13The variable is x 5x – 2 + 2 = 13 + 2We are multiplying it by 5, and subtracting 2 First, undo the subtracting by adding 2. 5x = 15Then, undo the multiplication by dividing by 5. 5 5 x = 3**Suppose there are variables on both sides of the equation.**The trick now, is to get the variables on the same side by adding or subtracting them. Solve for x in the equation 4x + 5 = x – 4 We have two terms with the variable, 4x and x.4x + 5 = x - 4 We’ll move the variable4x – x = x – x - 4 with the smaller3x + 5 = -4 coefficient, x. To do this we have to look at the sign in front of the variable we’re moving. Since the is no sign we know it is +. To move this Variable we do the opposite, so we’’ll subtract x from both sides.**Now we proceed as before:3x + 5 = -43x + 5 – 5 = -4 –**5Subtract 5 from both sides.3x = -933 Divide both sides by 3. x = -3**With any math there are new vocabulary words and rules we**must follow. Let’s look at some of the new terms and rules before we move on.**Solving Equations by Adding or Subtracting**Equation – a mathematical sentence that shows two expressions are equal. Solve – to find the answer or solution. Solution – the value that makes an equation true. Inverse operations – operations that “undo” each other; addition and subtraction, multiplication and division. Isolate the variable – to get the variable on one side of an equation or inequality by itself in order to solve. Open sentence – an equation that contains at least one variable.**Addition Property of Equality – states you can add the**same amount to both sides of an equation and the equation remains true.2 + 3 = 52 + 3 + 4 = 5 + 4 9 = 9 ? trueSubtraction Property of Equality – states you can subtract the same amount from both sides of an equation and the equation remains true.4 + 7 = 114 + 7 – 3 = 11 – 38 = 8 ? true**Addition and subtraction are inverseoperations, which means**they “undo” each other. To solve an equation, use inverse operations to isolate the variable, or to get the variable on one side of the equal sign by itself. x + 4 = 9 subtract 4 from both sidesx + 4 – 4 = 9 – 4 Subtraction property of equalityx + 0 = 5 Identity Property of Zero: x + 0 = 5check:x + 4 = 95 + 4 = 9 substitute 5 for x 9 = 9 ? true**w – 3 = 9Add 3 to both sides**w – 3 + 3 = 9 + 3 Addition Property of Equality w + 0 = 12 Identity Property of Zero: w + 0 = wcheck:12 – 3 = 9 Substitute 12 for w 9 = 9 ? TrueIt is very important to write all the steps and check your answer each time you solve an equation.**Solving Equations by Multiplication or Division**Multiplication Property of Equality – states you can multiply the same amount on both sides of an equation and the equation remains true. 4 · 3 = 12 2 · 4 · 3 = 12 · 2 24 = 24 Division Property of Equality – states you can divide the same amount on both sides of an equation and the equation remains true. 4 · 3 = 12 4 · 3 = 12 22 12 = 6 2**Multiplication and Division are inverseoperations, which**means they “undo” each other. To solve an equation, use inverse operations to isolate the variable, or get the variable on one side of the equal sign by itself. 7x = 35 Divide both sides by 7. 7x = 35 Division Property of Equality 7 7 1x = 5 1 · x = x X = 5 Check: 7x = 35 7 (5) = 35 substitute 5 for x 35 = 35 ? true**n ÷ 5 = 7 Multiply both sides by 5n**÷ 5 · 5 = 7 · 5 Multiplication Property of Equality n = 35check: n ÷ 5 = 735 ÷ 5 = 7 Substitute 35 for n 7 = 7 ? TrueIt is very important to write all the steps and check your solution each time you solve an equation.**Sometimes it is necessary to solve equations by using 2 or**more inverse operations. For instance, the equation 6x – 2 = 10.Always start with the operation that is the farthest away from the variable. 6x – 2 = 10 Add 2 to both sides first. 6x – 2 + 2 = 10 + 2 Addition Property of Equality 6x = 12Divide both sides by 6 66 Division Property of Equality x = 2 Check: 6x – 2 = 10 6(2) – 2 = 10 Substitute 2 for x 12 – 2 = 10 10 = 10 ? true**m + 15 = 25**50 = h – 3 4d = 144 x/3 = 18 S + 2 = 13 4x + 3 =19 y/2 – 5 = 1 26 = 3f + 10f 4(2x -1) + 3x = 11 144 = 12h Solving equationsGet you pencil and calculator ready and try these problems.**Evaluating and solving simple expressions and equations,**using order ofoperations, and using variables to solve real-world problems is the first step to becoming “good” at math. These skills lay the foundation for studies of algebra, geometry, and statistics.**Using Formulas**Formulas are equations used to show relationships between quantities.**Using Formulas (equations)**A formula or equation shows the relationship among certain quantities. The formula below can be used to find the miles per gallon achieved by a car. number of miles ÷ # of equals miles per driven gallons gas gallon m÷ g = mpg You drove 294 miles before stopping to get gas. Your gas tank holds 12 gallons of gas. What gas mileage does your car get? 294 ÷ 12 = 24.5 mpg**The formula was distance traveled by a moving object is d =**rt, where d represents distance in kilometers (km), r represents the rate in kilometers per hour (km/h), t represents the time in hours (h). • Use the formula d = rt to find the indicated variables. • r = 60 km/h; t = 4 h; d = • d = 100 km; t = 2 h; r = • r = 55 km/h; d = 110 km; t = • r = 35 km/h; t = 3 h; d = • d = 210 km; t = 7 h; r = • r = 80 km/h; d = 320 km; t =**The formula I = prt is used to find the amount of simple**interest on a given amount, where I is the interest; p is the principal amount; r is the rate of percent; and t is the time in years. Thurman borrowed $13,500 from his brother for 4 years at an annual percentage rate of 6%. How much interest will he pay if he pays the entire loan off at the end of the fourth year? What is the total amount he will repay?**Formulas are used everyday to solve problems, whether you**are computing gas mileage for your car (mpg = m ÷ g) or changing degrees Celsius to Fahrenheit (F = 9/5C + 32), or even solving the Pythagorean Theorem (a² + b² = c²) to find distance.