CHAPTER 1.3 Solving Equations

The equations are equivalent If they have the same solution(s) Equivalent Equations The equations are equivalent If they have the same solution(s)

Example: Determine whether 4x = 12 and 10x = 30 are equivalent equations Determine whether 3x = 4x and 3/x = 4/x are equivalent equations

ADDITION AND MULTIPLICATION PROPERTIES OF EQUALITY Let a, b, and c represent algebraic expressions Addition property of equality: If a = b, then a + c = b + c then a –c = b - c Multiplication property of equality: If a = b, then a(c) = b (c)

APPLYING THE ADDITION PROPERTIE OF EQUALITY In each equation, the goal is to isolate the variable on one side of the equation. To accomplish this, we use the fact that the sum of a number and its opposite is zero and the difference of a number and itself is zero. p – 4 = 11 To isolate p, add 4 to both sides (-4 +4 = 0). p – 4 +4 = 11 +4 p- + 0 = 15 p = 15 Simplify CHECK

Let’s try: TRUE Using the addition principle, adding 4.7 to both sides The solution of this equation is 18.6 Substituting 18.6 for y TRUE

Applying the Multiplication Properties of Equality Tip: Recall that the product of a number and its reciprocal is 1. For example: To obtain a coefficient of 1 for the x-term, divide both sides by 12 12x = 60 12x = 60 12 12 Simplify x = 5 Check!

Tip: When applying the multiplication or division properties of equality to obtain a coefficient of 1 for the variable term, we will generally use the following convention: If the coefficient of the variable term is expressed as a fraction, we usually multiply both sides by its reciprocal. If the coefficient of the variable term is an integer or decimal, we divide both sides by the coefficient itself. Example: To obtain a coefficient of 1 for the q-term, multiply by the reciprocal of which is Simplify. The product of a number and its reciprocal is 1. CHECK!

Let’s try: Using the multiplication principle, multiply by 1/8 to both sides The solution of this equation is 9 Substituting 9 for t TRUE

Steps to solve a linear Equation in One Variable Simplify both sides of the equation. Clear parentheses Combine like terms Use the addition property of equality to collect the variable terms on one side of the equation. Use the addition property of equality to collect the constant terms on the other side of the equation. Use the multiplication property of equality to make the coefficient of the variable term equal to 1. Check your answer.

We will want to find those values of x that make the equation true by isolating the x (this means get the x all by itself on one side of the equal sign) Since the x is in more than one place and inside of parenthesis the first thing we’ll do is get rid of parenthesis by distributing. Now let’s get all constants (terms without x’s) on the right side. We’ll do this by adding 8 to both sides. + 8 + 8 + 3 We are ready to get all x terms on the left side by adding 10x to both sides. + 10x + 10x Now get the x by itself by getting rid of the 12. 12x means 12 times x so we get rid of it by dividing both sides by 12. 12 12

Let’s check this answer by substituting it into the original equation to see if we get a true statement. ? Distribute and multiply ? Distribute ? Get a common denominator ? It checks!

Let’s try: Using the addition principle, adding (-3x) to both sides Using the multiplication principle, multiply by 1/4 to both sides TRUE

Conditional Equations, Identities, and Contradictions An equation that is true for some values of the variable but false for other values. Contradictions Equation with no solution Identities An equation that has all real numbers as its solutions.

Classify the equation Using the addition principle Simplifying Divide both sides b -1. There is one solution, -2. For other choices of x, the equation is false. This equation is conditional since it can be true or false, depending on the replacement for x.

Classify the equation Using the distributive law Combining like terms. Using the addition principle, adding -3x to both sides The equation is false regardless of what x is replaced with, so all real numbers are solutions. There is no solution. This equation is a contradiction.

Classify the equation Using the distributive law Combining like terms. The equation is true regardless of what x is replaced with, so all real numbers are solutions. This equation is a Identity.

Let’s try: