Math 20-1

Chapter 8/9 Systems of Equations and Solving

Inequalities

General Outcome: Develop algebraic and graphical reasoning through the study of relations.

Specific Outcomes: RF6. Solve, algebraically and graphically, problems that involve systems

of linear-quadratic and quadratic-quadratic equations in two variables.

RF7. Solve problems that involve linear and quadratic inequalities in two variables. [C, PS, T, V]

[ICT: C6–4.1, C6–4.3]

RF8. Solve problems that involve quadratic inequalities in one variable. [CN, PS, V]

Mark Assignments

8.1: Page 435– 438 # 1-5, 8, 10, 11, 15

8.2: Page 451 – 456 # (1, 2, 3ace, 4)(6, 8, 9, 10, 12, 17, 24)

Quiz 8 Date:

Mark Assignments

9.1: Page 472–473 # 1ac, 2bd, 3, 4, 5, 6(without

technology), 7, 9, 11, 12.

9.2: Page 484 – 487 # 1, 2, 3ac, 4 – 9[choose 2], 10, 11

9.3: Page 496 – 500 # 1 – 2(choose 2), 3, 5, 6, 8, 9, 10.

Quiz 9 Date:

Chapter 8/9 Test Date:

8.1 – Solving Systems of Equations Graphically

KEY IDEAS:

System of Equations:

Two or more equations involving the same variables being considered at the same time.

System of Linear-Quadratic Equations:

A linear equation and a quadratic equation involving the same variables (being

considered at the same time).

The graph of the system involves a line and a parabola.

Example:

System of Quadratic-Quadratic Equations:

Two quadratic equations involving the same variables (being considered at the same

time).

The graph of the system involves two parabolas.

Example:

Solution to a System of Equations:

A set of values for the variables (any ordered pair (x, y)) that satisfy all equations in the

system.

Example: The point (2, 4) is a solution of the system

y = x + 2

y = x2

Number of Solutions:

A system of linear-quadratic or quadratic-quadratic equations may have no solution, one real

solution, or two real solutions. A quadratic-quadratic system of equations may also have an

infinite number of real solutions.

Examples:

Example 1 – Relate a System of Equations to a Context

1. Blythe Hartley, of Edmonton, Alberta, is one of Canada’s best springboard divers. She is

doing training dives from a 3-m springboard. Her coach uses video analysis to plot her

height above the water.

a) Which system could represent the scenario? Explain your choice and why other graphs do

not model this situation?

b) Interpret the point(s) of intersection in the system your chose.

2. Two divers start their dives at the same time. One diver jumps from a 1-m springboard and

the other jumps from 3-m spring board. Their heights above the water are plotted over time.

a) Which system could represent the scenario? Explain your choice and why other graphs

do not model this situation?

b) Explain why there is no point of intersection in the graph your chose.

Example 2 – Solving a System of Linear-Quadratic Equations Graphically

a) Solve the following system of equations graphically:

4𝑥 – 𝑦 + 3 = 0

2𝑥2 + 8𝑥 – 𝑦 + 3 = 0

b) Verify your solution.

Example 3 – Solving a System of Quadratic-Quadratic Equations Graphically

a) Solve the following system of equations graphically:

2𝑥2 – 16𝑥 – 𝑦 = −35

2𝑥2 – 8𝑥 – 𝑦 = −11

b) Verify your solution.

Example 4 – Apply a System of Linear-Quadratic Equations

Engineers use vertical curves to improve the comfort and safety of roadways. Vertical curves are

parabolic in shape and are used for transitions from one straight grade to another. Each grade

line is tangent to the curve.

There are several vertical curves on the Trans-Canada Highway through the Rocky Mountains.

To construct a vertical curve, surveyors lay out a grid system and mark the location for the

beginning of the curve and the end of the curve.

Suppose surveyors model the first grade line for a section of road with the linear equation 𝑦 =

−0.06𝑥 + 2.6, the second grade line with the linear equation 𝑦 = 0.09𝑥 + 2.35, and the

parabolic curve with the quadratic equation 𝑦 = 0.0045𝑥2 + 2.8.

a) Write the two systems of equations that would be used to determine the coordinates of

the points of tangency.

b) Determine the points of tangency graphically, to the nearest hundredth.

c) Interpret the point of tangency.

8.2 – Solving Systems of Equations Algebraically

Review:

Quadratic-Quadratic and Linear-Quadratic Systems of Equations can have (depending on

the type of system), _______________________ real solutions.

A solution to a system of equations is any set of values for the variables (any ordered pair

(x, y)) that satisfy all equations in the system.

Example: The point (2, 4) is a solution of the system

y = x + 2

y = x2

We can solve systems of equations graphically.

We can also use algebraic methods of __________________________ or

________________________ learned last year.

Solving By Substitution:

- Isolate one variable in one equation.

- Substitute the expression into the other equation and solve for the remaining

variable.

- Substitute the value(s) into one of the original equations to determine the

corresponding value(s) of the other variable.

- Verify your answer by substituting into both original equations.

Example 1 – Solve by Substitution

a) Solve the following system of equation algebraically using substitution.

b) Verify your solution.

1. 5𝑥 − 𝑦 = 10

𝑥2 + 𝑥 − 2𝑦 = 0

2. 6𝑥2 − 𝑥 − 𝑦 = −1

4𝑥2 − 4𝑥 − 𝑦 = −6

Solving By Elimination:

- If necessary, rearrange the equations so that the like terms align.

- If necessary, multiply one or both equations by a constant to create equivalent

equations with a pair of variables with the same or opposite coefficients.

- Add or subtract to eliminate one variable and solve for the remaining variable.

- Substitute the value(s) into one of the original equations to determine the

corresponding value(s) of the other variables.

- Verify your answer(s) by substituting into both original equations.

Example 2 – Solve by Elimination

a) Solve the following system of equation algebraically using elimination.

b) Verify your solution.

1. 3𝑥 + 𝑦 = −9

4𝑥2 − 𝑥 + 𝑦 = −9

2. 2𝑡2 − 4𝑡 + ℎ = 3

4𝑡 − 2ℎ + 7 = 0

We’ve seen systems of equations with 1 solution and 2 solutions.

What about systems of equations with NO SOLUTIONS?

𝑦 = −2𝑥2 + 1 𝑛 − 𝑚2 = 7

4𝑥 + 3𝑦 = 12 2𝑚2 − 2𝑛 = −1

What about systems of equations with INFINITELY MANY SOLUTIONS?

𝑦 = −2(𝑥 + 4)2 − 5

𝑦 = −2𝑥2 − 16𝑥 − 37

Example 1 – Number Problems

Determine two integers that have the following relationships: Fourteen more than twice the first

integer gives the second integer. The second integer increased by one is the square of the first

integer.

Example 2 – A Canadian cargo plane drops a crate of emergency supplies to aid-workers on

the ground. The crate drops freely at first before a parachute opens to bring the crate gently to

the ground. The crate’s height, h, in metres, above the ground t seconds after leaving the aircraft

is given by the following two equations.

ℎ = −4.9𝑡2 + 700 represents the height of the crate during the free fall.

ℎ = −5𝑡 + 650 represents the height of the crate with the parachute open.

a) How long after the crate leaves the aircraft does the parachute open? Express your

answer to the nearest hundredth of a second.

b) What height above the ground is the crate when the parachute opens? Express your

answer to the nearest metre.

c) Verify your solution.

9.1 – Linear Inequalities in Two Variables

KEY IDEAS:

A linear inequality in two variables may be in one of the following four forms:

_____________________________

_____________________________

_____________________________

_____________________________

An inequality in two variables (x and y) describes a region on the Cartesian plane. The

ordered pair (x, y) is a solution to a linear inequality if the inequality is true when the

values of x and y are substituted into the inequality.

The solution region (solution set) is all the points in the Cartesian plane that satisfy

the inequality.

The boundary is the line or curve that separates the Cartesian plane into two regions.

This line may, or may not be a part of the solution region.

The boundary is drawn as a solid line and included in the solution

region if the inequality is ≥ or ≤. The boundary is drawn as a dotted

line and not included in the

solution region if the inequality is

< or >.

In the following example, Ax + By = C is the

boundary. For one solution region, Ax + By > C

is true. For the other solution region, Ax + By < C

is true.

Use a test point to determine which region is the

solution region for the inequality.

Example 1 – Graph a Linear Inequality of the Form Ax + By ≤ C

a) Graph 2𝑥 + 3𝑦 ≤ 6.

b) Determine if the point (-2, 4) is a part of the solution.

Example 2 – Graph a Linear Inequality of the Form Ax + By > C

Graph 10𝑥 − 5𝑦 > 0.

Example 3 – Write an Inequality Given Its Graph

Write an inequality to represent the following graph.

Example 4 – Write and Solve an Inequality

Suppose that you are constructing a tabletop using aluminum

and glass. The most you can spend on material is $50. Glass

costs $60/m2 and aluminum costs $1.75/ft. You can choose

the dimensions of the table and the amount of material used.

Find all possible combinations of materials sufficient to make

the tabletop.

9.2 – Quadratic Inequalities in One Variable

KEY IDEAS:

A quadratic inequality in ONE variables may be in one of the following four forms:

_____________________________

_____________________________

_____________________________

_____________________________

Quadratic inequalities can be solved graphically or algebraically. The solution set to a

quadratic inequality in one variable can have ______________, ______________

or ______________________.

Inequalities in ONE VARIABLE are different than inequalities in TWO

VARIABLES.

Inequalities in TWO variables - the solution is a shaded region on the

graph representing options for both x and y that will satisfy the

inequality.

Inequalities in ONE variable – the solution is a section of the graph

representing all the options for x that will satisfy the inequality.

Example 1 – Solve a Quadratic Inequality of the Form ax2 + bx + c ≤ 0, a > 0

Solve 𝑥2 − 2𝑥 − 3 ≤ 0.

Method 1: Graph the Corresponding Function

1. Graph the function:

2. To determine the solution to 𝑥2 − 2𝑥 − 3 ≤ 0, look for the values of x for which the

graph of f(x) ___________________________________.

Method 2: Finding Roots (Zeros) and Test Points

1. Solve the related equation to determine the roots (zeros).

2. Then use a number line and test points to determine the intervals that satisfy the

inequality.

Method 3: Case Analysis

1. Factor the inequality.

2. Determine the possible cases when the product of these two factors satisfies the

inequality (is negative or positive).

Method 4: Sign Analysis

1. Factor the quadratic expression.

2. Determine when each of the factors is positive, negative or zero and plot each one

on a number line.

3. Combine the number lines to determine the solution to the inequality.

Example 2 – Use Sign Analysis to Solve a Quadratic Inequality

Solve −𝑥2 + 3𝑥 + 10 > 0 using sign analysis.

Example 3 – Use Case Analysis to Solve a Quadratic Inequality

Solve −𝑥2 + 𝑥 + 12 < 0 using case analysis.

HINT: ALWAYS MAKE YOUR

x2 TERM POSITIVE BY

REARRANGING YOUR

EQUATION!!!!! THIS MAKES

SIGN AND CASE ANALYSIS

MUCH EASIER!!

Example 4 – Use Roots and Test Points to Solve an Inequality

Solve 2𝑥2 − 7𝑥 > 12 using roots and test points.

Example 5 – Applying Quadratic Inequalities

If a baseball is thrown at an initial speed of 15 m/s from a height of 2 m above the

ground, the inequality −4.9𝑡2 + 15𝑡 > −2 models the time, t, in seconds, that the

baseball is in flight. During what time interval is the baseball in flight? (Use graphing to

solve.)

9.3 – Quadratic Inequalities in Two Variables

KEY IDEAS:

A quadratic inequality in two variables may be in one of the following four forms:

_____________________________

_____________________________

_____________________________

_____________________________

A quadratic inequality in two variables represents a region of the Cartesian plane with a

_____________________ as the boundary.

The graph of a quadratic inequality is the set of points (x, y) that are solutions to the

inequality.

Example 1 – Graph a Quadratic Inequality in Two Variables

c) Graph 𝑦 < −2(𝑥 − 3)2 + 1.

d) Determine if the point (2,-4) is a solution to the inequality.

Example 2 – Solve a Quadratic Inequality in Two Variables

Solve 𝑦 ≥ 𝑥2 − 4𝑥 − 5.

Example 3 – Determine a Quadratic Inequality that Define a Solution Region

You can use a parabolic reflector to focus sound, light, or radio waves to a single point. A

parabolic microphone has a parabolic reflector attached that directs incoming sounds to the

microphone. Rene, journalist is using a parabolic microphone as he covers the Summer Festival

in Vancouver. Write a quadratic inequality to describe the region that Rene can cover his

microphone if the reflector has a width of 50 cm and a maximum depth of 15 cm.

Example 4 – Write and Solve an Inequality

Manila rope is used from rappelling down a cliff when alpine rock climbing. Manila

rope can safely support a mass, M, in pounds, modelled by the inequality M ≤ 1450d2,

where d is the diameter of the rope, in inches. Graph the inequality to examine how the

mass that the rope supports is related to the diameter of the rope.