7. Algebra Flashcards
What is a term?
One or more numbers and/or letters all connected by multiplication or division.
What is an Algebraic expression?
An algebraic expression is one more terms in a phrase. It’s not a whole “sentence,” so an expression doesn’t include an equal sign. here’s an example: 4x+2y+38c.
What is an equation?
An equation is a mathematical statement that states that two mathematical expressions are equal. An equation always includes an equal sign, like this example: 7a + 3b = 588.
What is a variable?
A variable is a number in disguise. In this expression, x and y are the variables: 7x + 10y = 27. The are called variables because the numbers they represent can vary from problem to problem. When you write your own equations, you can use any letters that make sense to you.
If the same letter appears more than once in a given equation or expression, it stands for the same number in all instances in that equation/expression. In 3x +2x =10, the first x doesn’t represent a different number from the second x. In this case, x = 2 (both times).
What is a coefficient?
A coefficient is the number part of an algebraic term. In this equation, 8 and 12 are the coefficients: 8a -12b = -4.
What is a constant?
A constant is a term in an algebraic expression that contains only a number; it’s a term without a variable attached to it. In this equation, 5 is the constant:14d +12a +5.
What are real numbers?
Real number are actual quantities, like amounts, ages, distances, and temperatures. A real number can be an integer, fraction, or decimal, and it can be rational or irrational. They’re numbers that appear on a number line, like this:
What are rational numbers?
Rational numbers are the set of real numbers, fractions, and terminating decimals. Rational numbers can always be written as the ratio or quotient of two integers. These are examples of rational numbers: 1/2, 5, 0.25.
What are irrational numbers?
Irrational numbers can’t be expressed as a quotient or ration of two integers. With irrational numbers, the decimal form is nonrepeating and nonterminating, like pi. You can’t write pi as a quotient of two integers, and its decimal form goes on forever. The same goes for square roots and other roots.
What are exponents? & What should you know about exponents?
An exponent tells you how many times to use a value (its base ) in multiplication. In this equation, 2 and x are both exponents: s2 + 3x = 470.
- Any base raised to the power of one equals itself: x1 = x.
- Any base raised to the zero power (except 0) equals 1: x0=1
- To multiply terms with the same base, add the exponents: x2 (x3) = x2+3 = x5
- If a base has a negative exponent, it’s equal to its reciprocal (inverse) with a positive exponent: x-3 = 1/x3
- When a product has an exponent, each factor is raised to that power: (xy)3 = x3y3.
What is a polynomial?
A polynomial is an expression that contains one or more terms, like constants, variables, and exponents. The exponents on the variables in a polynomial are always whole numbers. Here’s an example of a polynomial of a polynomial: 3ab - 2x2 + 2y3 - 7. Other special names for polynomials signal whether they have one, two, or three terms. For example, 2ab7 is a monomial because it has one term. 6x - 3 is a binomial because it has two terms, and 4a + 7b3 is a trinomial because it has three terms..
What are like terms?
In algebra, you combine like terms — terms that have matching variables in the expression or equation. The variables must have the same power to be like terms. These are all like terms: 4xy, 7xy, 32xy. Because xy is the same, you can combine them if you’re working with an equation, like this:
4xy + 7xy + 32xy = 215
43xy = 215
Why should you do the same thing on what you did on either side of an equation?
Because this proportional calculation keeps the equation equal.
Can you combine like terms?
If like terms are on the same side of the equation, yes!
When you transpose a term on the opposite side of the equal sign to find x, you always have to use an inverse operation.
True or false
True
How do you solve a multistep equation?
(equations with two same variables on the opposite side e.g., 3x + 3 = 9 + x)
INVERSE OPERATIONS!
How to simplify an expression?
- Remove whatever parentheses you can by multiplying factors
- Use exponent rules to remove any remaining parentheses (if applicable).
- Combine like terms
- Combine constants
- 3(2+x)+4(4x+3)-(x2)2
- 6+3x+16x+3-x4
- 6+19x+12-x4
- 19x+18-x4
Create an equation for the following problems
- Kathy sold 10 more carnival tickets than Bonnie did. Together, they sold 480 tickets. How many tickets did Bonnie sell?
- t + (t + 10) = 480
You can perform any calculation on either side of an equation as long as…
you do the same thing to both sides of the equation.
How do you solve one-step equations involving addition and subtraction?
- x + 47,372 = 50,000
- Isolate x on one side of the equal sign (x = 50,000 - 47,372).
Why do you get a negative number when you divide or multiply by:
negative and positive?
and get a positive number when you divide or multiply by:
negative and negative?
To balance each side of the equation.
One variable is always equal to one times that variable.
True or False
True
Describe the FOIL method.
Foil is a technique for distributing two binomials in algebra. FOIL stands for First, Outer, Inner, and Last, and it refers to problems like this one: (x+2) (x+5).
What are two-variable equations?
Two-variable equations are what you see when working with graphs and in some other instances, they are usually multiple solutions (x + 3 = y).
When you have two or more of these equations and they all have exponents of 1, the equations are called linear systems, and to solve them, you either have to use substitution or combine equations.
What are linear systems and how do you solve them?
Linear systems are two or more “two-variable equations” in which you can solve by using the following techniques:
a. substitution
or
b. combining equations
How to perform the substitution to solve a linear system?
Solve the following linear system.
x+3=y
3x+y=7
When a system is simple, like this one, substitution is the way to go.
x+3=y
3x+y=7
In this system, you know that y=x+3. Plug x+3 into the second equation, in place of y, to find the value of x, and the simplify and solve.
3x+x+3=7
4x+3=7
4x+3-3=7-3
4x=4
4x/4 = 4/4
x=1
because x = 1, substitute that value into the original equation, y=x+3, to determine that y=4.
How to perform the combining equations when solving a more complicated linear system?
Solve the following linear system.
12x-9y=37
8x+9y=23
When a system is more complicated, like this one, you need to combine them.
12x-9y=37
8x+9y=23
You have to add these two equations to figure out just one of the variables, like this:
12x-9y=37
+ 8x+9y=23
20x =60
x=60/20
x=3
When you solve for x, you see that x=3. Plug 3 in for the value of x in either of the original equations to figure out the value of y.
- 8(3)+9y=23
- 24+9y=23
- 9y=-1
- y=-1/9
Combining equations may require you to change one equation entirely to figure out the answer. Here’s another system as an example:
7x+10y=36
2x-y=-9
To solve by combining, multiply the bottom equation by 10. The equation becomes 20x-10y=-90. Add it to the first equation to eliminate y. The result is 27x=-54. Divide to find the value of x:
27x/27 = -54/27
x=-2
Take the value of x, which is -2, and substitute it into one of the original e3quations to solve for y (I use the second):
- 2x-y=9
- 2(-2)-y=9
- -4-y=9-4
- y=5
What is factoring?
Factoring is the process that you do in order to find the original numbers that were multiplied together to produce that product.
What is a greatest common factor?
The greatest common factor is the highest number that evenly divides all the terms in the expression and the lowest power of each variable.
How do you factor a two-term equation?
Factor this equation:
4xy + 2x2
- Find the greatest common factor. [look at both the constants (numbers) and variables. In this case, the highest number that divides into 4 and 2 is 2. And the highest variable that divides into both xy and x2 is x. You can see that the greatest common factor is 2x).
- Divide both terms in the expression by the greatest common factor. [When you have a fraction, such as 4xy+2x2/2x, the resulting terms are 2y+x).
- Multiply the entire expression (From Step 2) by the greatest common factor (From Step 1) to set the expression equal to its original value. [Doing so produces 2x(2y + x).
How do you factor a three-term equation (x2 + bx + c)?
Factor this equation:
x2 - 12x + 20
- Find the factors of the first term of the trinomial. [The factors of the first term, x2, are x and x (x * x = x2). Put those factors (x and x) on the left side of two sets of parentheses: (x )(x ).
- Determine whether the parentheses will contain positive or negative sign. [You can see that the last term in the trinomial (+20) has a plus sign. that means the signs in the parentheses must be either both plus signs or both minus signs. Because the second term (-12x) is a negative number, both factors must be negative: (x - )(x - ).
- Find the two numbers that go into the right sides of the parentheses. [This part can be tricky. The factors of the third term, when added together or subtracted, must equal the second term of the trinomial. In this example, the third term is 20 and the second term is -12x. you need to find the factors of 20 (the third term) and -2 + -10 = -12 (the second term). Plug in these numbers: (x - 2)(x - 10). Thus, the factors of x2 - 12x + 20 are (x - 2) and (x - 10).
What is a quadratic equation?
A quadratic equation is an equation that includes the square of a variable. The exponent in these equations is never higher than 2 (because it would then no longer be the square of an unknown but a cube or something else). Here are some of quadratic equations:
- x2 - 4x = -4
- 2x2 = x + 6
- x2 = 36
How do you solve a quadratic equation?
Solve the following quadratic equation:
7y2 = 28
7y2 = 28
- Get rid of the 7 by dividing both sides by 7. (y2 = 4).
- Using the square root rule, take the square root of both sides of the equation. You know that √y2 = y and √4 = ±2, so y = ±2.
How do you solve a complex quadratic equation?
solve the ff. equation:
x2 - 2x = 15.
x2 - 2x = 15
- Put all the terms on one side of the equal sign, making the equation equal to zero. (ax2 + bx + c = 0). ||| x2 - 2x - 15 = 0
- Factor the equation. ||| (x - 5)(x + 3) = 0
- Split the equation in two, setting each factor equal to zero. ||| x - 5 = 0 to x = 5 or x + 3 = 0 to x = -3.
- The solutions are x = 5 or x = -3.
What are inequalities?
Inequalities are algebra problems which states that two quantities aren’t equal to each other; thus, they are inequalities.
How do you solve inequalities?
To solve inequalities, you follow the same rules you would for solving an equation. The only special rule for inequalities takes effect when you multiply or divide both sides of the inequality by a negative number. In that case, the inequality sign is reversed.
|||
5x + 7 < 22
5x < 15
5x/5 < 15/5
x > 3
