CHAPTER 2.1. ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION PROPERTIES OF EQUALITY. Determine whether the given number is a solution to the equation. 4x + 7 = 5. x = -½. Substitute -½ for x. Simplify.
ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION PROPERTIES OF EQUALITY
4x + 7 = 5
x = -½
Substitute -½ for x.
Right-hand side equals the left=hand side. Thus, -½ is a solution to the equation 4x + 7 = 5
-6x + 14 = 4
x = 3
Substitute 3 for x.
Right-hand side does not equal the left=hand side. Thus, 3 is not a solution to the equation -6x + 14 = 4
A linear equation in one variable is equivalent to an equation of the form:
If we have a linear equation we can “manipulate” it to get it in this form. We just need to make sure that whatever we do preserves the equality (keeps both sides =)
We can add or subtract the same thing from both sides of the equation.
Notice this is the equation above where a = 3 and b = -3.
While this is in the general form for a linear equation, we often want to find all values of x so that the equation is true. You could probably do this one in your head and see that when x = 1 we’d have a true statement 0 = 0
then a + c = b + c
then a – c = b -c
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 = 4 +4
p- + 0 = 15
p = 15
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
x = 5
To obtain a coefficient of 1 for the q-term, multiply by the reciprocal of
Simplify. The product of a number and its reciprocal is 1.
The quotient of a number and 4 is 6
The product of a number and 4 is 6
Negative twelve is equal to the sum of -5 and a number
The value 1.4 subtracted from a number is 5.7