1. Systems of equations Vision 1
Prepared September 2010 by C. Cichanowicz

2. y = ax + b (function form)
Equation of a Line
y = ax + b (function form)
Graphing: No matter what form the equation is given in (function or general form), manipulate the equation so that it is in this form
a – the rate of change (or slope in the graph)
b – the initial value
Prepared September 2010 by C. Cichanowicz

3. Equation of a Line The slope
In a graph: it is the steepness of the line (if the line goes down, the slope is negative...if the line goes up, the slope is positive)
In a word problem: it is the value at which something increases or decreases. (Ex. A salary of 9$/hr, decreases by 10% daily)
Prepared September 2010 by C. Cichanowicz

4. Equation of a Line Slope (rate of change):
Prepared September 2010 by C. Cichanowicz

5. Equation of a Line Determine the slope
Prepared September 2010 by C. Cichanowicz

6. Equation of a Line The initial value
In a graph: it is where the line crosses the y-axis
In a word problem: it is the starting value before the rate is taken into consideration.
Ex. A base salary before commission
An employee makes 200$ plus 50$/computer sold
Initial value
Slope or
rate of change
Prepared September 2010 by C. Cichanowicz

7. Equation of a Line Determine the equation of the line from the graph
Prepared September 2010 by C. Cichanowicz

8. Equation of a Line Determine the equation of the line from the graph
Prepared September 2010 by C. Cichanowicz

9. Equation of a Line Graphing equations
From the equation of the line determine the initial value, this will be your starting point on the y-axis
From the equation of the line establish the value of the slope
If given as a fraction, remember:
If given as a whole number, the
x-value is 1
+ go up
- go down
+ go right
- go left
Prepared September 2010 by C. Cichanowicz

10. Equation of a Line The form: Ax + By + C = 0
When an equation is given in this form transform it into the form y = ax + b
4x + 3y – 15 = 0
Isolate y (any operations you perform will be performed on each side of the equation
4x – 4x + 3y – = - 4x + 15
3y = - 4x + 15
Divide by the coefficient of y
Simplify:
Prepared September 2010 by C. Cichanowicz

11. Systems of Equations
Word problems can be turned into a system of equations...
In the form : y=ax+b
In the form: ax+by=c (this is usually some kind of cost equation)
Solving Systems: to determine intersection coordinates
Substitution
Comparison
Elimination
Prepared September 2010 by C. Cichanowicz

12. Systems of Equations The solution is (19, -25) Substitution
2x + y = 13
4x + 3y = 1
Transform one of the equations to function form
2x – 2x + y = 13 – 2x
y = 13 – 2x
Substitute the transformed equation into the other equation
4x + 3 (13 – 2x) = 1
4x + 39 – 6x = 1
-2x + 39 = 1
-2x = 1 -39
-2x = -38
-2x ÷ -2 = -38 ÷ -2
x = 19
To find y, substitute the value obtained for x into the transformed equation
y = 13 – 2x
y = 13 – 2(19)
y = 13 – 38
y = -25
The solution is (19, -25)
Prepared September 2010 by C. Cichanowicz

13. Systems of Equations Comparison The Solution is: (-2, 5) y = 2x + 9
Both equations are equal to y, we can therefore say they are equal to each other (solve by algebraic manipulation)
2x + 9 = -3x – 1
2x + 3x + 9 = -3x + 3x – 1
5x + 9 = -1
5x + 9 – 9 = -1 – 9
5x = -10
5x ÷ 5 = -10 ÷ 5
x = -2
To find y, substitute the value obtained for x into one of the equations
y = 2x + 9
y = 2(-2) + 9
y =
y = 5
The Solution is: (-2, 5)
Prepared September 2010 by C. Cichanowicz

14. Systems of Equations Elimination 2x + 5y = -4 3x -2y = 13
Make one of the variables the same
Subtract the 2 equations
To find y, substitute the value obtained for x into one of the equations
2x + 5y = -4
2x + 5(-2) = -4
2x – 10 = -4
2x – =
2x = 6
2x ÷ 2 = 6 ÷ 2
x = 3
The solution is: (3, -2)
Prepared September 2010 by C. Cichanowicz

15. Systems of Equations
Solve the following systems using the method of your choice
5x – 3y = 12
y = -2x + 18
2x + 3y = 6
5x + 10y = 20
y = -2x + 9
y = x - 3
Prepared September 2010 by C. Cichanowicz