Chapter 7 - XYZ Custom Plus

424Chapter 7 Solving Equations2. Is a = −3 the solution to theequation 6a − 3 = 2a + 4?Example 2Is a = −2 the solution to the equation 7a + 4 = 3a − 2?Solution When a = −2the equation 7a + 4 = 3a − 2becomes 7(−2) + 4 = 3(−2) − 2−14 + 4 = −6 − 2−10 = −8A false statementBecause the result is a false statement, we must conclude that a = −2 is not thesolution to the equation 7a + 4 = 3a − 2.BAddition Property of EqualityInstructor NoteAs you know, the addition propertyof equality can be extended to includesubtracting the same numberfrom both sides of an equation. Idon’t mention this in class, nor doI work examples using subtraction,because I have found that my studentswill make fewer mistakes ifthey think in terms of addition. However,in the next section we coverthe multiplication property of equality,and we do extend that propertyto include dividing both sides of anequation by the same number.We want to develop a process for solving equations with one variable. The mostimportant property needed for solving the equations in this section is called theaddition property of equality. The formal definition looks like this:Addition Property of EqualityLet A, B, and C represent algebraic expressions.If A = Bthen A + C = B + CIn words: Adding the same quantity to both sides of an equation never changesthe solution to the equation.This property is extremely useful in solving equations. Our goal in solving equationsis to isolate the variable on one side of the equation. We want to end upwith an equation of the formx = a numberTo do so we use the addition property of equality. Remember to follow this basicrule of algebra: Whatever is done to one side of an equation must be done to theother side in order to preserve the equality.3. Solve for x: x + 5 = −2NoteWith some of theequations in this section,you will be ableto see the solution just by lookingat the equation. But it is importantthat you show all the steps usedto solve the equations anyway. Theequations you come across in thefuture will not be as easy to solve,so you should learn the stepsinvolved very well.Example 3Solve for x: x + 4 = −2Solution We want to isolate x on one side of the equation. If we add −4 toboth sides, the left side will be x + 4 + (−4), which is x + 0 or just x.x + 4 = −2x + 4 + (−4) = −2 + (−4) Add −4 to both sidesx + 0 = −6Additionx = −6x + 0 = xThe solution is −6. We can check it if we want to by replacing x with −6 in theoriginal equation:Whenx = −6the equation x + 4 = −2becomes −6 + 4 = −2−2 = −2 A true statementAnswers2. No 3. −7