1. Solving Linear Equations with One Variable

2. Solving Linear Equations
To solve a linear equation in one variable:
1. Simplify both sides of the equation.
2. Use the addition and subtraction properties to get all variable terms on the left-hand side and all constant terms on the right-hand side.
3. Simplify both sides of the equation.
4. Divide both sides of the equation by the coefficient of the variable.
Example: Solve x + 1 = 3(x  5).
x + 1 = 3(x  5)
Original equation
x + 1 = 3x  15
Simplify right-hand side.
x = 3x  16
Subtract 1 from both sides.
 2x =  16
Subtract 3x from both sides.
x = 8
Divide both sides by 2.
The solution is 8.
Check the solution:
(8) + 1 = 3((8)  5)  9 = 3(3)
True
Solving Linear Equations

3. Using a Common Denominator
Equations with fractions can be simplified by multiplying both sides by a common denominator.
Example: Solve
The lowest common denominator of all fractions in the equation is 6. 6
Multiply everything on each side by 6.
3x + 4 = 2x + 8
Simplify.
3x = 2x + 4
Subtract 4.
x = 4
Subtract 2x. 4
Check.
True
Using a Common Denominator

4. Let the number of 10 cents coins in the coin purse = d.
Alice has a coin purse containing $5.40 in 10 cents coins and 20 cents coins. There are 30 coins all together. How many 10 cents coins are in the coin purse?
Let the number of 10 cents coins in the coin purse = d.
Then the number of 20 cents coins = 30  d.
10d + 20(30  d) = 540
Linear equation
10d  20d = 540
Simplify left-hand side.
10d  20d =  60
Subtract 480.
10d =  60
Simplify right-hand side.
d = 6
Divide by 10.
There are 6, 10 cents coins in Alice’s coin purse.
Example: Word Problem

5. Three consecutive integers can be represented as n, n + 1, n + 2.
The sum of three consecutive integers is 54. What are the three integers?
Three consecutive integers can be represented as n, n + 1, n + 2.
n + (n + 1) + (n + 2) = 54
Linear equation
3n + 3 = 54
Simplify left-hand side.
3n = 51
Subtract 3.
n = 17
Divide by 3.
The three consecutive integers are 17, 18, and 19.
= 54.
Check.
Example: Word Problem

6. x and y -intercepts
The x-intercept is the point where a line crosses the x-axis.
The general form of the x-intercept is (x, 0). The y-coordinate will always be zero.
The y-intercept is the point where a line crosses the y-axis.
The general form of the y-intercept is (0, y). The x-coordinate will always be zero.

7. To summarize…. To find the x-intercept, plug in 0 for y.
To find the y-intercept, plug in 0 for x.

8. Find the x and y- intercepts of x = 4y – 5
Plug in x = 0
x = 4y - 5
0 = 4y - 5
5 = 4y
= y
(0, ) is the
y-intercept
x-intercept:
Plug in y = 0
x = 4y - 5
x = 4(0) - 5
x = 0 - 5
x = -5
(-5, 0) is the
x-intercept

9. Example Question: Draw the graph of 2x + y = 4 Solution x = 0
1st Co-ordinate = (0,4)
y = 0
2x + 0 = 4
2x = 4
x = 2
2nd Co-ordinate = (2,0)

10. y x So the graph will look like this. 2x + y = 4 1 2 3 4 5 6 7 8 1 2 3 1 2 3 4 5 6 7 8
–7 –6 –5 –4 –3 –2 –1 -1 -2 -3 -4 -5 -6
2x + y = 4

11. Simultaneous Equations

12. What are they? Simply 2 equations
With 2 unknowns
Usually x and y
To SOLVE the equations means we find values of x and y that
Satisfy BOTH equations [work in]
At same time [simultaneously]

13. We have the same number of y’s in each
Elimination Method
We have the same number of y’s in each
2x – y = 1 A + B
3x + y = 9
If we ADD the equations, the y’s disappear 5x
= 10
Divide both sides by 5 x
= 2
2 x 2 – y = 1
Substitute x = 2 in equation A
4 – y = 1
Answer
x = 2, y = 3
y = 3

14. What if NOT same number of x’s or y’s?
3x + y = 10
If we multiply A by 2 we
get 2y in each B
5x + 2y = 17 A -
6x + 2y = 20 B
5x + 2y = 17 x
= 3
In B
5 x 3 + 2y = 17
Answer
x = 3, y = 1
15 + 2y = 17
y = 1

15. Linear Simultaneous Equationsy
Linear Simultaneous Equations
Substitution Example
2x - y = -1 1.
x + 2y = 7 2.
Sub x = 1 into 2.
1 + 2y = 7
Make ‘y’ the subject in 1.
2y = 6
y = 2x a.
y = 3
Substitute 1a into equation 2.
x + 2(2x + 1) = 7
5x + 2 = 7
Solution (1, 3)
5x = 5
x = 1

16. Slope of a Line
Slope of a line:
rise
run

17. Slope-Intercept Form The equation y = mx+b
is called the slope-intercept form of an equation of a line. The letter m represents the slope and b represents the y-intercept.

18. Point-Slope Form The point-slope form of the equation of a line is
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
Cross-multiply and substitute the more general x for x2

19. Example
Find the equation of the line through the points (–5, 7) and (4, 16).

20. Example
Find the equation of the line through the points (–5, 7) and (4, 16).
Solution:
Now use the point-slope form with m = 1 and (x1, x2) = (4, 16). (We could just as well have used (–5, 7)).

21. Distance between two points

22. The formula

23. Why?
The formula, derived from the Pythagoras theorem states that square a and square b, add them together, and square root them, you get the length of c, hypotenuse.

24. What is the distance between the points A(5, –1) and B(–4, 5)?
Example
Given the coordinates of two points we can use the formula
to directly find the distance between them. For example:
What is the distance between the points A(5, –1) and B(–4, 5)?
A(5, –1) B(–4, 5) x1 y1 x2 y2
Point out that it doesn’t matter which point is called (x1, y1) and which point is called (x2, y2).
It can help to write x1, y1, x2 and y2 above each coordinate as shown before substituting the values into the formula.
The answer in this example is written in surd form. An alternative would be to write it to a given number of decimal places; for example, (to two decimal places).

25. The Mid-Point of a Line
In general, the coordinates of the mid-point of the line segment joining (x1, y1) and (x2, y2) are given by:
(x2, y2)
(x1, y1) x y
is the mean of the x-coordinates.
Talk through the generalization of the result for any two points (x1, y1) and (x2, y2). As for the generalization for the distance between two points, it doesn’t matter which point is called (x1, y1) and which point is called (x2, y2).
is the mean of the y-coordinates.

26. Example:
Use this activity to explore the mid-points of given line segments.
Establish that the x-coordinate of the mid-point will be half way between the x-coordinates of the end points. This is the mean of the x-coordinates of the end-points.
The y-coordinate of the mid-point will be half way between the y-coordinates of the end points. This is the mean of the y-coordinates of the end-points.

27. END ANY QUESTION?