 # Section 2.1 Solving Linear Equations

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Section 2.1 Solving Linear Equations
Strategy for Solving Algebraic Equations: 1. Use the distributive property to remove parentheses:           becomes 3x – = 18 – 5x 2.  Combine like terms on either side of the equation.  -9 and 3 can be added to get -6. 3x – 6 = x 3.  Use the addition or subtraction properties of equality to get the variables on one side of the = symbol and the constant terms on the other. 3x and 5x are like terms. Add 5x to each side to get the variable terms on the left. 3x + 5x – 6 = 18 -5x + 5x   x - 6 = 18 4. Continue to combine like terms whenever possible.         6 and 21 are like terms.  Since 6 is subtracted from 8x, add 6 to both sides to move it to the other side.              8x =              8x = 24 5.  Undo the operations of multiplication and division to isolate the variable.             Divide both sides by 8 to get x by itself.             8x/8  = 24/8                 x = 3 6.  Check the results by substituting your found value for x into the original equation.         3(x - 3) + 3 = 18 – 5x 3(3-3) + 3 = 18 – 5(3) 3(0) + 3 = 18 – 15 3 = 3

5 + 3(a+4) = 7a – (9-10a) + 4 Now it’s your turn! Try:
Step 1: Use distributive property to remove parentheses Step 2: Combine like terms on each side. Step 3: Use addition property of equality to combine like terms between sides. Step 4: Continue to combine like terms wherever possible. Step 5: Once variable terms are are combined and isolated, use multiplication property of equality (multiply both sides by the reciprocal of the coefficient) to completely isolate the variable. State your conclusion: a = ____ Step 6: Check your solution in the original equation.

The Trick with Fractions!
Fractions are messy to deal with. When solving equations with fractions in them we can take advantage of the multiplication property of equality to get rid of them while keeping an equivalent equation. What we do is multiply BOTH SIDES of the equation (that is everything on each side) by the LCD of all the fractions.

The Trick with Decimals!
Decimals are also are messy to deal with. When solving equations with fractions in them we can take advantage of the multiplication property of equality to get rid of them while keeping an equivalent equation. What we do is multiply BOTH SIDES of the equation (that is everything on each side) by the power of 10 that corresponds to the number with the most decimal places. The number with the most decimal places is .05 (2 places). This corresponds to 102, or If we multiply both sides by 100, all the decimal numbers get changed to whole numbers.

Solve: k – 0.2(k + 5) = -.1

Not all equations have a solution
Not all equations have a solution. Sometimes it is impossible to find a value for a variable that would make the equation true. Example: Solve y = y + 2 Is there any number that you can add 2 to it and still get the same number? No. What happens when we try to solve it? Combine like terms. There’s a variable on each side, so to eliminate the y on the right, you’d have to subtract y from both sides. y = y + 2 -y y 0 = 0 + 2 0 = 2 ?? You see, if you try to solve an equation that is “unsolvable” you will get a false statement. This means that no matter what value you have for y, the equation will always be false.

Some equations have an infinite number of solutions.
Example: 3(x – 1) + 1 = 4x – (x + 2) Step 1: Use distributive property 3x – = 4x – x – 2 Step 2: Combine like terms on each side 3x – 2 = 3x – 2 Step 3: Combine like terms between sides using addition property of equality. 3x – 2 = 3x – 2 3x = 3x -3x -3x 0 = 0 !! This statement is true no matter what value of x you choose. Therefore the solution set is {all real numbers}

Section 2.2 Formulas and Functions
A formula or literal equation is an equation involving two or more variables. The variable that is isolated is called a “function” of the other variables. That is, it depends on other variables for its value. Examples: Example 1 p. 78 Solve for F. You want to isolate F. We can multiply both sides of the equaiton by 9/5 so we don’t have fractions when using the distributive property to remove the ( ). Now add 32 to both sides If the Celsius temperature is 35 degrees, what is the temperature in Fahrenheit?

Example 2 p. 79 Solve 3a – 2b = 6 for a Isolate a by getting everything that doesn’t have an a on the other side.

Example 3 p. 80