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In this video, we're going to study Properties of Equality. The first Property of Equality that we're going to consider
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is the Addition Property of Equality. It says that if a, b, and c are real numbers and a is equal to b, then a plus c is equal
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to b plus c. This means that I can add the same number to both sides of this equal sign
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and still have a true statement. Because subtraction can also be written as an
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addition statement, plus a negative number, this property holds true for subtraction as well.
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And I can write that if a equals b then a minus c is equal to b minus c.
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Let's look at these examples. The directions are to solve the equation.
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To solve an equation, we're going to isolate the variable on one side of the equal sign
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Using the Addition Property of Equality, I can rewrite this equation to an equivalent
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equation that has the same set of solutions as this original equation. Notice that on the
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right side we have x plus one-half. To isolate the x, I need to get rid of the plus one-half.
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So we will subtract one-half from both sides of the equal sign. On the right-hand side, I can see that plus one-half
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minus one-half would be zero. So I'm left with x plus zero, or simply x.
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And on the left-hand side, we have three-fourths minus one-half. Now remember, to add or subtract
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fractions, you must have a common denominator. The lowest common denominator of four and two
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is four. So I need to multiply this fraction by two over two. That does not change the value
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because I'm simply multiplying by one. This now gives me two over four, so I have three
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over four minus two over four, leaving us with one over four. So my solution is x equals one-fourth.
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This solution is the solution to the original equation. The next example we have is x minus one point two is equal to
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three point five. Again, to solve this equation, I want x isolated on one side of the equal sign. So this minus
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one point two needs to move. I'm going to add one point two to both sides of this equation.
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By the Addition Property of Equality, when I add the same thing to both sides of the equation,
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I don't change the solutions. And so negative one point two plus one point two is zero.
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x plus or minus zero is just x. And on the right-hand side, three point five plus one
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point two is four point seven. So our solution to this equation is x equals four point seven.
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Our next example is seven a minus five equals eight a. Again, we are trying to solve
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the equation which means we want to isolate the variable on one side of the equal sign.
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It does not matter whether I isolate the variable on the left or on the right. In this case,
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I'm going to choose to isolate the variable on the right side as a slightly more efficient method of
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solving this equation to reduce the number of operations that I have. So if I want to
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move the seven a to the right side, I am going to subtract seven a from both sides of this equation.
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Notice that seven a minus seven a is zero and zero minus five is negative five. So on the left,
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we have negative five. On the right, we have eight a minus seven a which is one a, or just a.
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This is an equivalent equation to the equation that was given. It's in a more convenient form because
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now we know exactly the value of a. a is negative five. The solution to the original equation is negative five.
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In our next example, we have one-fourth m plus seven-fourths m equals m plus five-sixths.
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We are solving this equation. So we want to have all of the m terms on one side of the equation.
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First however, I'm going to notice that I can combine these terms here using
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the Distributive Property. This sum can be rewritten as one-fourth plus seven-fourths
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times m. And that is equal to m plus five-sixths. One-fourth plus seven-fourths is eight-fourths,
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and eight-fourths is equivalent to two. So this left side is two m. The right side is m plus five-sixths.
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Now, I'm going to move the m from the right side to the left by subtracting m from both sides.
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On the right side, m minus m is zero, and zero plus five-sixths is five-sixths.
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And on the left side, two m minus m is m. So we have m equals five-sixths. This is the solution to the original equation.
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The Multiplication Property of Equality states that if a, b, and c are real numbers,
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c is not equal to zero, and a is equal to b, then a times c is equal to b times c.
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Because division can also be rewritten as a multiplication problem, this property holds true
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for division as well as long as c is not equal to zero. If a is equal to b, then a divided by c is equal to b divided by c.
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We'll now use this property to solve the following equations. The first equation is two-thirds x equals five.
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We want to isolate x on one side of the equal sign. Currently, it is being multiplied times two-thirds. I can divide
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both sides of this equation by two-thirds, or I can also multiply both sides of this equation by the reciprocal of two-thirds.
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The reciprocal of two-thirds is three-halves. So we will multiply both sides of this equation
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by three-halves. When you multiply a number times its reciprocal, the product is one. Notice
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six divided by six would be one. So the left side of this equation is one x, or just x.
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The right side of this equation is five times three-halves which is fifteen-halves.
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So the solution to our original equation, two-thirds x equals five, is fifteen-halves. In our next equation, we have point four
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x equals point eight eight. The point four is being multiplied times x,
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so we will divide both sides by point four. So that we will get x remaining on the left side.
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Dividing both sides of this equation by point four. On the left side, point four divided by point four is one,
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and one x is simply x. And on the right-side, point eight eight divided by point four is two point two.
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So the solution to our equation, point four x equals point eight eight, is two point two.
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In some problems, it is necessary to use both the Addition Property of Equality and the
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Multiplication Property of Equality to solve the equation. Let's look at these next few examples.
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We're going to solve each equation. In the first example, we have seven-eighths a
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is equal to eight-sevenths a. We're going to begin by using the Addition Property of Equality
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to get both terms with a on the left side of the equation. So we're going to subtract eight-sevenths a from both sides.
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On the right side of the equation, we have eight-sevenths a minus eight-sevenths a which is zero.
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On the left side of the equation, we have seven-eighths a minus eight-sevenths a.
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I’m going to rewrite this subtraction using the Distributive Property.
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So this becomes seven-eighths minus eight-sevenths a is equal to zero.
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Now, I could subtract these fractions and come up with an equivalent answer for that difference,
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and then divide both sides by that number. But first let's think about this problem. I'm going to want to isolate a on the left,
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so I need to divide by this number, whatever this number is. So if I divide both sides of this equation
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by this number, I’m going to have zero divided by some number. Zero divided by a non-zero number
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is just zero. So this divided on both sides is still going to give me zero. The solution
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to the equation seven-eighths a equals eight-sevenths a is zero.
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In the next example, we have six-fifths y minus five is equal to negative seven. We want to solve
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the equation, meaning that we want to isolate the variable y on one side of the equal sign.
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So we can see that we have a six-fifths multiplied times y, and we also have a minus five
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added to this term or positive five subtracted from this term. We're going to begin by moving
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the minus five. We do that by adding five to both sides of the equation.
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On the left-hand side, I can see that minus five plus five is zero, leaving us with six-fifths y.
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And on the right-hand side of the equation, we have minus seven plus five which is a negative
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two. Now I have six-fifths multiplied times y. We'll use the Multiplication Property of Equality
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and multiply both sides of this equation by the reciprocal of six-fifths. The reciprocal of
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six-fifths is five-sixths. So we will multiply both sides of the equation by five-sixths.
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Five-sixths times six-fifths is one, leaving us with one y, or just y, on the left.
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Negative two times five-sixths gives us negative five-thirds on the right. The solution to this equation is negative five-thirds.
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In our last example, we have fifteen x minus nine is equal to five x plus nineteen.
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We want to solve the equation, so we want all of the variables on one side of the equation.
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So I'm going to begin by subtracting five x from both sides of the equation,
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essentially moving this term to the left side. I notice that I’m also going to want to move this
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minus nine because it does not contain an x. I want to move that to the right side
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of the equation, so I will also add nine to both sides of this equation.
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Here I have minus fifteen x minus five x, excuse me I have fifteen x minus five x.
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I'm going to use the Distributive Property to write this as fifteen minus five
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times x. We have minus nine plus nine which is zero equals five x minus five x is zero,
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and nineteen plus nine is twenty-eight. Fifteen minus five is ten, so on the left side I have ten
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x. And on the right side, I have twenty-eight. Now using the Multiplication Property of Equality
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which also holds for division, I'm going to divide both sides of this equation by ten.
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Ten divided by ten is one, leaving me with one x, or just x. And twenty-eight
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divided by ten can be simplified by dividing both the numerator and denominator by two,
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leaving us with fourteen-fifths. The solution to this original equation is fourteen-fifths.
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All of the equations that we have looked at and solved in this section have been linear equations
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in one variable. Let's look at the definition of a linear equation in one variable. A linear
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equation in one variable is an equation that is written in the form ax plus b equals c where a, b,
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and c are real numbers and a is not equal to zero. Notice a cannot be equal to zero here
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because then zero times x would be zero, and you would no longer have a variable in the equation,
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or a linear equation in one variable can be an equation that can be written
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in this form using properties of equality or properties of arithmetic. Properties of
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equality include the Multiplication Property of Equality and the Addition Property of Equality.
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The properties of arithmetic that we have studied include the Commutative Property,
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the Associative Property, and the Distributive Property. The form of this equation a x plus b equals c is a linear equation
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in one variable. Notice that the variable x here has an exponent of one. We don't see the exponent of one.
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It's implied. So any equation that has x to any other power besides one is not a linear equation in one
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variable. If the x is underneath of a radical sign or in the denominator of a fraction, that equation
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also is not a linear equation. ax plus b equals c is the form of a linear equation in one variable.