Category Archives: Math 10

• Write both equations in standard form.

• Make the coefficients of one variable opposites.

• Add the equations to eliminate one variable.

• Solve for the remaining variable.

• Substitute the solution into one of original equations.

• Write the solution as an ordered pair.

• Check that the ordered pair is a solution to both original equations.

Here are clues to know when a word problem requires you to write a system of linear equations:

(i) There are two different quantities involved: for instance, the number of adults and the number of children, the number of large boxes and the number of small boxes, etc.

(ii) There is a value associated with each quantity: for instance, the price of an adult ticket or a children’s ticket, or the number of items in a large box as opposed to a small box.

-Such problems often require you to write two different linear equations in two variables. Typically, one equation will relate the number of quantities (people or boxes) and the other equation will relate the values (price of tickets or number of items in the boxes).

Here are some steps to follow:

1. Understand the problem.

Understand all the words used in stating the problem.

Understand what you are asked to find.

Familiarize the problem situation.

2. Translate the problem to an equation.

Assign a variable (or variables) to represent the unknown.

Clearly state what the variable represents.

3. Carry out the plan and solve the problem.

Use substitution , elimination or graphing method to solve the problem.

Example:

The cost of admission to a popular music concert was $162 for 12 children and 3 adults. The admission was $122 for 8 children and 3 adults in another music concert. How much was the admission for each child and adult?

1 . Understand the problem:

The admission cost for 12 children and 3 adults was $162.

The admission cost for 8 children and 3 adults was $122.

2. Translate the problem to an equation.

Let x represent the admission cost for each child.

Let y represent the admission cost for each adult.

The admission cost for 12 children plus 3 adults is equal to $162.

That is, 12x+3y=162.

The admission cost for 8 children plus 3 adults is equal to $122.

That is, 8x+3y=122.

33 . Carry out the plan and solve the problem.

Subtract the second equation from the first.

Substitute 10 for x in 8x+3y=122.

Therefore, the cost of admission for each child is $10 and each adult is $14.

By graphing a set of linear equations, you can determine whether the system has no solution, one solution, or an unlimited number of solutions.

A system has one solution when (m1) is not equal to (m2).

Ex.

2x+5=y 

-2x+2=y

A system has no solutions when (m1) is equal to (m2) and (b1) is not equal to (b2)

Ex.

2x+5=y

2x-5=y

A system as and infinite number of solutions when (m1) is equal to (m2) and (b1) is equal to (b2)

ex

2x-3=y

2x-3=y

There are three main forms of linear equations.

Slope-intercept (y=mx+b), Point-slope (y−y1​=m(x−x1​)), and Standard (Ax+By=C)

Slope-Intercept: where m is slope and b is the y-intercept

Point slope: where m is slope and (x1​,y1​) is a point on the line

Standard: where A, B, and C are constants

b is the slope of the line. Slope means that a unit change in x, the independent variable will result in a change in y by the amount of b.

slope = change in y/change in x = rise/run

Slope shows both steepness and direction. With positive slope the line moves upward when going from left to right. With negative slope the line moves down when going from left to right.

Calculating the slope of a linear function

Slope measures the rate of change in the dependent variable as the independent variable changes. Mathematicians and economists often use the Greek capital letter D or Δ as the symbol for change. Slope shows the change in y or the change on the vertical axis versus the change in x or the change on the horizontal axis. It can be measured as the ratio of any two values of y versus any two values of x.

Slope types:

Positive Slope

Negative Slope

Undefined Slope

Zero Slope

The graph of a function f is the set of all points in the plane of the form (x, f(x)). We could also define the graph of f to be the graph of the equation y = f(x). So, the graph of a function if a special case of the graph of an equation.

Vertical Line Test

The graph of the equation y² = x + 5 is shown below

By the vertical line test, this graph is not the graph of a function, because there are many vertical lines that hit it more than once. Think of the vertical line test this way. The points on the graph of a function f have the form (x, f(x)), so once you know the first coordinate, the second is determined. Therefore, there cannot be two points on the graph of a function with the same first coordinate. All the points on a vertical line have the same first coordinate, so if a vertical line hits a graph twice, then there are two points on the graph with the same first coordinate. If that happens, the graph is not the graph of a function.

In mathematics, a function can be defined as a rule that relates every element in one set, called the domain, to exactly one element in another set, called the range. For example, y = x + 3 and y = x² – 1 are functions because every x-value produces a different y-value.

A relation is any set of ordered-pair numbers. In other words, we can define a relation as a bunch of ordered pairs.

How to Determine if a Relation is a Function?

• Examine the x or input values.
• Examine also the y or output values.
• If all the input values are different, then the relation becomes a function, and if the values are repeated, the relation is not a function.

Are the following ordered pairs are functions?

1. W= {(1, 2), (2, 3), (3, 4), (4, 5)}
2. Y = {(1, 6), (2, 5), (1, 9), (4, 3)}

1. All the first values in W = {(1, 2), (2, 3), (3, 4), (4, 5)} are not repeated, therefore, this is a function.
2. Y = {(1, 6), (2, 5), (1, 9), (4, 3)} is not a function because, the first value 1 has been repeated twice.

The domain and range are defined for a relation and they are the sets of all the x-coordinates and all the y-coordinates of ordered pairs respectively. For example, if the relation is, R = {(1, 2), (2, 2), (3, 3), (4, 3)}, then:

Domain = the set of all x-coordinates = {1, 2, 3, 4}

Range = the set of all y-coordinates = {2, 3}

A domain of a function refers to “all the values” that go into a function. The domain of a function is the set of all possible inputs for the function.

The range of a function is the set of all its outputs.

A point in a plane contains two components where order matters! It comes in the form ($x$,y) where $x$ comes first, and $y$ comes second.

• The $x$-value tells how the point moves either to the right or left along the $x$-axis.
• The $y$-value tells how the point moves either up or down along the $y$-axis.

A coordinate plane is composed of two lines intersecting at a 90-degree-angle (making them perpendicular lines) at the point (0,0) known as the origin.

• The $x$-component of the point ($x$,$y$) moves the point along a horizontal line. If the $x$-value is positive, the point moves “$x$-units” towards the right side. On the other hand, if the $x$-value is negative, the point moves “$x$-units” towards the left.
• The $y$-component of the point ($x$,$y$) moves the point along a vertical line. If the $y$-value is positive, the point moves “$y$-units” in an upward direction. However, if the $y$-value is negative, the point moves “$y$-units” in a downward direction.

The intersection of the $x$-axis and $y$-axis results to the creation of four (4) sections or divisions

• The first quadrant is located at the top right section of the plane.

• The second quadrant is located at the top left section of the plane.

• The third quadrant is located at the bottom left section of the plane.

• The fourth quadrant is located at the bottom right section of the plane.

the exponent of a number tells us how many times we should multiply the base.

For example, 8², 8 is the base, and 2 is the exponent.

We know that 8² = 8 × 8.

A negative exponent tells us, how many times we have to multiply the reciprocal of the base.

Consider the 8^-2, here, the base is 8 and we have a negative exponent (-2).

8^-2 is expressed as 1/8² = 1/8×1/8.

MORE NEGATIVE EXPONENT EXAMPLES –

5^-4= 1/5⁴

x^-6= 1/x⁶

1/y^-5= y⁵

b^-6/d^-5 = d⁵/b⁶

8^-9= 1/8⁹

40^-4= 1/40⁴

2250^-2= 1/2250²