PREC11 Unit3 notes - PRE-CALCULUS 11 Unit 3 Day 1 SOLVING...

PRE-CALCULUS 11 Unit 3 – Day 1: SOLVING SYSTEMS OF EQUATIONS GRAPHICALLY
SYSTEMS OF EQUATIONS
A system of equation is a collection of two or more equations with the same variables.
• A system of equations with two variables will usually have two equations.
To solve any system of equations, find all the values of the variables that satisfies every
equation in the system .
• The solution of a system with equations having only two variables x and y are all the
ordered pairs that satisfy all the equations .
The solution of a system of equations can be found:
• graphically , or
• algebraically - either by the substitution method or the elimination method.
SOLVING SYSTEMS OF EQUATIONS BY GRAPHING
The coordinates of every point on a graph must satisfy its equation; therefore the
point(s) that are on every graph must have the solution(s) of the system.
To solve a system of equations graphically:
• Graph every equation on the same coordinate plane.
• Find all the points that are on all the graphs; look at the point(s) of intersection.
example: Solve this system of equations graphically.
y = x
2
and y = x + 2
o
Graph each equation.
o
Look at the points of intersection, (2,4)
and ( - 1,1); are they on all the graphs?
Answer: x = - 1 and y = 1 or x = 2 and y = 4 check!
y
x
0
-4
4
-4 4

Unit 3: Day 1 notes - Solving Systems of Equations Graphically Page 2
of 2
exercise: Solve graphically; round to one decimal place if necessary.
a) y = x
2
- 4 x + 2 b) y = ( x + 1)
2
- 4
3 x + 2 y - 11 = 0 y = - 2 x
2
+ 7
x = - 1 and y = 7 or x ≈ 3.5 and y ≈ 0.3 x ≈ - 2.2 and y ≈ - 2.6 or x ≈ 1.5 and y ≈ 2.4
exercise: How many solutions can a linear-quadratic system have?
exercise: How many solutions can a system with two quadratic functions have?
y
x
0
- 4
4
-4
4
y
x
0
- 4
4
-4
4

PRE-CALCULUS 11 Unit 3 – Day 2: SOLVING SYSTEMS OF EQUATIONS ALGEBRAICALLY (Part 1)
SOLVING SYSTEMS ALGEBRAICALLY
The solution of a system of equations can be solved:
• graphically, or
• algebraically - either:
o with the substitution method , or
o with the elimination method .
SOLVING SYSTEMS OF EQUATIONS WITH THE SUBSTITUTION METHOD
To solve a system of equations algebraically using The Substitution Method:
• Solve one of the equations for one of the variables; choose carefully.
• Take the expression equal to that variable and substitute it into the other equation;
the result should be a single equation with a single variable.
• Solve this equation; find the roots - the values of this first variable.
• Substitute each of these roots into an equation with both variables - one at a time;
each of these roots will produce an equation with the second variable.
• Solve these equations; find the value of the second variable.
example: Solve this system of equations graphically. x
2
- y = 0 and x - y = - 2
o
Solve for y in the linear function. y = x + 2
o
Substitute
x
+ 2 into the quadratic function. x
2
- ( x + 2) = 0
o
Solve this quadratic equation. x
2
- x - 2 = 0
( x + 1) ( x - 2) = 0
x = - 1 or x = 2
o
Substitute each of these x -values into the
linear function. y = - 1 + 2 or y = 2 + 2
y = 1 or y = 4
Answer: x = - 1 and y = 1 or x = 2 and y = 4 check!