Algebra

Question 1

which sign is used in linear equations and in linear inequalities?

Answer 1

linear equation “=”
linear inequalities “>, <, =>, =<”

Question 2

When solving linear equations, what do we have to keep in mind?

Answer 2

the equation has to remain equivalent to the original equation, so we always treat both sides equally: whenever we do something to one side, we must do the exact same thing to the other side.

Question 3

When solving linear inequalities, what do we have to keep in mind about signs?

Answer 3

If the coefficient of [x] is positive, the inequality sign maintains its direction when we divide by the coefficient to isolate [x].
If the coefficient of [x] is negative, we must reverse the direction of the inequality sign when we divide by the coefficient to isolate [x].

Question 4

How can we determine the number of solutions for a linear equation

Answer 4

If the equation can be rewritten in the form [x=a], where [a] is a constant, then that equation has one solution.
If the variable can be eliminated from the equation, and what remains is the equation [a=b], where [a] and [b] are different constants, then the equation has no solution.
If the equation can be rewritten in the form [x=x], then the equation has infinitely many solutions. (No matter what the value of [x] is, it will always equal itself!)

Question 5

what is the equation for the slope?

Answer 5

change in y/change in x
or (y2-y1)/(x2-x1)

Question 6

What are the two famous forms in which a linear relationship is written? And what does each one of them tell us about?

Answer 6

1)The slope-intercept form of a linear function, y=mx+b, where m and b are constants, tells us both the slope and the y-intercept of the line.
2)The standard form of a linear function, Ay+Bx=C, where, A, B, and C are constants, will often be used in word problem scenarios that have two inputs, instead of an input and an output.

Question 7

We can write the equation of a line, as long as we have, at least, what?

Answer 7

The slope of the line and a point on the line or two points on the line

Question 8

what is magnitude?

Answer 8

distance from 0.
ex: -3 has a greater magnitude than 2

Question 9

what are the characteristics of parallel lines in the xy-plane
what are the characteristics of perpendicular lines in the xy-plane?

Answer 9

Parallel lines in the xy-plane have the same slope.
Perpendicular lines in the xy-plane have slopes that are negative reciprocals of each other.

Question 10

What are systems of linear equations?

Answer 10

A system of linear equations is usually a set of two linear equations with two variables.

Question 11

How does substitution work? (Systems of linear equations)

Answer 11

1) Isolate one of the two variables in one of the equations.
2) Substitute the expression that is equal to the isolated variable from Step 1 into the other equation. This should result in a linear equation with only one variable.
3) Solve the linear equation for the remaining variable.
4) Use the solution of Step 3 to calculate the value of the other variable in the system by using one of the original equations.

Question 12

How does elimination work? (Systems of linear equations)

Answer 12

1) Identify a pair of terms in the system that have both the same variable and coefficients with the same magnitude (ex: 2x and 2x, or 3y and -3y). If necessary, rewrite one or both equations so that a pair of terms have both the same variable and coefficients with the same magnitude.
2) Add or subtract the two equations in the system to eliminate the terms identified in Step 1. This should result in a linear equation with only one variable.
3) Solve the linear equation to obtain a value for the variable.
4)Now that you have figured out the value of one variable, plug that value into either equation to find the value of the other variable.

Question 13

How to determine the number of solutions a system of linear equations ?

Answer 13

y=mx+b
1) Rewrite both equations in slope-intercept form.
2) Compare the m- and b-values of the equations to determine the number of solutions.
3) If the two equations have different m-values, then the system has one solution.
4) If the two equations have the same m-value but different b-values, then the system has no solution.
5) If the two equations have both the same m-value and the same b-value, then the system has infinitely many solutions.

Question 14

What are systems of linear equations word problems?

Answer 14

Systems of linear equations word problems ask us to translate real-world scenarios into a system of two linear equations with two variables. Often, we’ll also be asked to solve the system.

Question 15

How to solve a system of linear equations word problem ?

Answer 15

1) Select variables to represent the unknown quantities.
2) Using the given information, write a system of two linear equations relating the two variables.
3) Solve the system of linear equations using either substitution or elimination.

Question 16

What are linear inequality word problems and system of linear inequalities?
What is an example of system of linear inequalities

Answer 16

While a linear equation gives us exactly one value when solved, a linear inequality gives us multiple values. A system of linear inequalities is just like a system of linear equations, except it is composed of inequalities instead of equations.
ex: You’re buying snacks for a party; you want to buy enough so that you don’t run out (snacks >= what people will eat), but you also don’t want to overspend (money spent on snacks =< budget for the party). If you manage to buy enough snacks without breaking your budget, you’ve solved a system of inequalities!

Question 17

What do these phrases translate to in a linear inequality word problems?
1)”More than [c]”, “greater than [c]”, or “higher than [c]” [>c]
2)”Less than [c]” or “lower than [c]”
3)”Greater than or equal to [c]” or “at least [c]”
4)”Less than or equal to [c]” or “at most [c]”
5)”No less than [c]”
6)”No more than [c]”
7)”Least”, “lowest”, or “minimum”
8)”Greatest”, “highest”, or 9)”maximum”
10)”A possible” value

Answer 17

1) >c
2) <c
3) >=c
4) <=c
5) >=
6) <=c
7) The smallest value that satisfies the inequality
8) The largest value that satisfies the inequality
9) Any value that satisfies the inequality

Question 18

What are graphs of linear systems and inequalities problems?

Answer 18

Graphs of linear systems and inequality problems deal with graphs of the following in the [xy]-plane:
-systems of linear equations
-linear inequalities
-systems of linear inequalities

Question 19

How do we know if we have to shade above the line or below the line in a linear inequality?
what about points on the line?

Answer 19

If [y] is greater than [mx+b], shade above the line.
If [y] is less than [mx+b], shade below the line.
Greater than ([>]) or less than ([<]) signs do not include points on the line in the solution set. We use a dashed line to show that the points on the line are not included.
Greater than or equal to >= or less than or equal to <= signs do include points on the line in the solution set. We use a solid line to show that points on the line are included.