Graphs of Linear Functions
Graphs of Linear Functions
Slide: 1
Co-ordinate System Y-axis
Graphs are drawn on what is known as the Cartesian coordinate system. System uses two perpendicular lines. The partitions created on the grid by the lines are known as quadrants. Going counterclockwise from the top right, they are quadrant I, quadrant II, quadrant III, and quadrant IV.
6 5
Quadrant II
3
X: negative Y: positive
X: positive Y: positive
2 1
−9 −8 −7
−6
−5 −4
−3 −2 −1 0 −1
1
2
3
4
5
6
7
−2
Quadrant III X: negative Y: negative
Slide: 2
Quadrant I
4
−3 −4 −5 −6
Quadrant IV X: potitive Y: negative
8
9
X-axis
Co-ordinate System We use ordered pairs to plot points on the graph.
Y-axis 6
(−6,5)
5
Example
(4 , 3) x-value
4
2
y-value
1 −9 −8 −7
Example
(−6 , 5)
(4,3)
3
−6
−5 −4
−3 −2 −1 0 −1
1
2
3
4
5
6
7
8
−2
(−8,−2)
−3 −4
Example
(−8 , −2)
Example
(7 , −4)
Slide: 3
−5 −6
(7,−4)
9
X-axis
Write the co-ordinates of the points shown on the graph. Y-axis
A
6 5
A:
4
E
3 2
B:
C:
D −9 −8 −7
−6
−5 −4
1 −3 −2 −1 0 −1
E:
Slide: 4
1
2
3
−2 −3
D:
C
−4 −5 −6
B
4
5
6
7
8
9
X-axis
Co-ordinate System Y-axis
You can determine the scale of your axes based on the points you need to plot. Example
(–4 , 250)
300 250
(–4,250)
200 150 100 50
−9 −8 −7
Note: The scales of the x and y axes do not need to be the same!
−6
−5 −4
−3 −2 −1 −50 −100 −150 −200 −250 −300
Slide: 5
1
2
3
4
5
6
7
8
9
X-axis
Linear Equations Linear equations consist of only constants and terms containing a single variable raised to the power of 1. Non-linear equations contain at least one term with more than one variable or a variable raised to a power greater than 1.
Linear
Non Linear
3x + 4y = 12
a2 + b2 = c2
a + 2b – 3c = 4d
x2 – 6x = –8
x=5 Slide: 6
a(b – 1) = c
Standard Form of Linear Equations The standard form of a linear equation with 1 variable, x, is as follows:
ax + b = 0 b x =– a
There is a single solution to this equation which can be determined by solving for x.
The standard form of a linear equation with 2 variables, x and y, is as follows:
a x + by + c = 0 Slide: 7
There are infinite pairs of x and y values which will satisfy this equation.
Graphing Linear Equations using a Table of Values Example
Rearrange the following linear equation with 2 variables for y:
6x + 5y = 20 5y = –6x + 20 6 y=– x+4 5
x
y
–5
10
0
4
5
–2
Select values of x that are divisible by 5. In this form, we can easily set up a table of x and y values which satisfy the equation.
Slide: 8
We call x an independent variable and y a dependent variable because the value of y depends on the value we select for x.
Graphing Linear Equations using a Table of Values We can represent the set of all solutions to the linear equation as a line on the Cartesian co-ordinate system.
6x + 5y = 20 Y 12 10 8 6
x
y
–5
10
0
4
5
–2
4 2 X −9 −8 −7
−6
−5 −4
−3 −2 −1 −2 −4 −6 −8 −10 −12
Slide: 9
1
2
3
4
5
6
7
8
9
Step 1: Choose a scale. Step 2: Plot the points. Step 3: Connect the points.
Graph the following linear equation using a table of values.
0 = –3x + 4y + 12
Y 6 5 4 3 2 1 X −9 −8 −7
−6
−5 −4
−3 −2 −1 −1 −2
x
y
−3 −4 −5 −6
Slide: 10
1
2
3
4
5
6
7
8
9
How do you represent the following linear equation on the Cartesian co-ordinate system?
x=4
Y
6
a.
5 4
c.
3 2
d.
1 −9 −8 −7
−6
−5 −4
−3 −2 −1 −1
1
2
3
X 4
5
−2 −3 −4 −5 −6
Slide: 11
b.
6
7
8
9
Intercepts The x-intercept of a linear equation is the point on the graph at which the line passes through the X axis. The y-intercept of a linear equation is the point on the graph at which the line passes through the Y axis. All 2-variable linear equations will have both an x- and y-intercept.
Y
Example
6 5
x-intercept: (4, 0)
4
y-intercept: (0, –3)
2
3
1 X −9 −8 −7
Given a 2-variable linear equation, you can solve for the x-intercept by setting y to 0, and solve for the y-intercept by setting x to 0, and solving the equation. Slide: 12
−6
−5 −4
−3 −2 −1 −1 −2 −3 −4 −5
−6
1
2
3
4
5
6
7
8
9
What are the x- and y- intercepts of the following linear equation?
14x – 4y – 28 = 0 a. x-intercept: (0, –7) y-intercept: (2, 0) b. x-intercept: (–7, 0) y-intercept: (0, 2) c. x-intercept: (0, 2) y-intercept: (–7, 0) d. x-intercept: (2, 0) y-intercept: (0, –7) Slide: 13
What are the x- and y- intercepts of the following linear equation?
Slide: 14
Graphing Linear Equations using Intercepts We can also graph a linear equation by drawing a line that connects the x- and y- intercepts.
Y 12
10 8 6 4 2 X −9 −8 −7
−6
−5 −4
−3 −2 −1 −2
1
2
3
4
5
6
7
8
9
−4 −6 −8
−10 −12
x-intercept: (2, 0) Slide: 15
y-intercept: (0, –7)
Slope-Intercept Form of Linear Equations We have already seen how a 2-variable linear equation in standard form can be rearranged to solve for y. This is called slope-intercept form.
Standard Form:
Slope-Intercept Form:
–3x + 4y + 12 = 0
3 y= x–3 4
Slide: 16
Slope-Intercept Form of Linear Equations We have already seen how a 2-variable linear equation in standard form can be rearranged to solve for y. This is called slope-intercept form.
y = mx + b b is the y-intercept of the line, and m is the slope of the line. The slope refers to the steepness of the line.
Given two points, (x1, y1) and (x2, y2), you can determine the slope of the line using the following formula: Slide: 17
𝑟𝑖𝑠𝑒 𝑦2 − 𝑦1 𝑚= = 𝑟𝑢𝑛 𝑥2 − 𝑥1
Determine the slope of the line containing the following 2 points:
(8, –2), (–2, 6)
Slide: 18
Slope of a Line The sign of m determines the direction of the line.
Positive m:
Negative m:
Y
Y
X
X
y increases
y decreases
As x increases…
Slide: 19
As x increases…
Slope of a Line The greater the absolute value of m, the steeper the line.
m=
𝟏 𝟑
m=3
Y
Y
X
Slide: 20
X
What is the equation of the following line? We have 2 points: (0, –7), (2, 0)
Y 12 10
Determining m:
8
𝑦2 − 𝑦1 0 − (−7) 7 𝑚= = = 𝑥2 − 𝑥1 2−0 2
6 4 2
X −9 −8 −7
Determining b:
−6
−5 −4
−3 −2 −1 −2
rise = 7
The y-intercept is (0, –7).
−4 −6 −8 −10 −12
Slide: 21
run = 2
7 y = x – 7 2
1
2
3
4
5
6
7
8
9
Find the equation of the line that is plotted in the graph shown. Y-axis 11
a. y = 2x
10 9 8
1 7 b. y = 2 x + 2 7 1 c. y = x + 2 2
7 6 5
4 3 2 1 −3 −2 −1 0 −1 −2
d. y = 2x + 6
1
2
3
4
5
6
7
8
9 10 11 12
X-axis
Find the equation of the line that is plotted in the graph shown. Y-axis 11 10 9 8
7 6 5
4 3 2 1 −3 −2 −1 0 −1 −2
1
2
3
4
5
6
7
8
9 10 11 12
X-axis
Parallel Lines Parallel lines have the same slope.
10 9 8
4 5
y= x+6
7
6 5
4 5
y= x+1
4 3 2 1 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 −2
Slide: 24
1
2
3
4
5
6
7
8
These lines are parallel
Perpendicular Lines The product of the slopes of perpendicular lines is –1.
10 9 8
y=
7 6
3 2
4 3 2 1
−2
Slide: 25
+4
y=− x+6
5
0 −8 −7 −6 −5 −4 −3 −2 −1 −1
2 x 3
1
2
3
4
5
6
7
8
These lines are perpendicular because their slopes are negative reciprocals
Write the equation of the line perpendicular to x − 2y = 12 that passes through the point (4,−4).
a. b. c. d.
y = −2x + 4 y = 2x − 12 y = −2x − 4 y = 2x + 12
Write the equation of the line perpendicular to x − 2y = 12 that passes through the point (4,−4).
Systems of Equations Two or more equations analyzed together are called systems of equations.
A system of equations that has one or many solutions is called a consistent system. A system of equations that has no solution is called an inconsistent system.
11 10 9 8 7 6 5 4
If a system of equations has many solutions then the equations are dependent. If a system of equations has one solution then the equations are independent.
Slide: 28
3 2 1 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 −2
1
2
3
4
5
6
7
8
Systems of Equations Lines may be non-parallel and intersecting, parallel and distinct, or parallel and coincident.
Non-Parallel and Intersecting Lines Slopes of lines are different. y-intercepts of lines may or may not be the same.
The system has one solution (the point where the lines intersect). The system is consistent. The equations are independent.
10 9 8 7
One Solution
6 5 4 3 2 1 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 −2
Slide: 29
1
2
3
4
5
6
7
8
Systems of Equations Lines may be non-parallel and intersecting, parallel and distinct, or parallel and coincident.
Parallel and Distinct Lines Slopes of lines are the same. 10
y-intercepts of lines are different.
The system has no solution.
No Solution
9 8 7
The system is inconsistent.
6 5
The equations are independent.
4 3 2 1 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 −2
Slide: 30
1
2
3
4
5
6
7
8
Systems of Equations Lines may be non-parallel and intersecting, parallel and distinct, or parallel and coincident.
Parallel and Coincident Lines Slopes of lines are the same. 10
y-intercepts of lines are the same.
The system has infinitely many solutions.
Many Solutions
9 8 7
The system is consistent.
6 5
The equations are dependent.
4 3 2 1 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 −2
Slide: 31
1
2
3
4
5
6
7
8
Solve this system of equations by graphing and classifying the system as consistent or inconsistent and the equations as dependent or independent. y=x+3 Equation 1 y = −x + 5 Equation 2
a. b. c. d.
(1,4); consistent; independent. (1,4); inconsistent; dependent. (4,1); consistent; independent. (1,4); inconsistent; dependent.
Solve this system of equations by graphing and classifying the system as consistent or inconsistent and the equations as dependent or independent. y=x+3 Equation 1 y = −x + 5 Equation 2 11 10 9 8 7 6 5 4 3 2 1 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 −2
1
2
3
4
5
6
7
8
Solve this system of equations by graphing and classifying the system as consistent or inconsistent and the equations as dependent or independent. y = −3x − 1 Equation 1 y = −3x + 7 Equation 2
a. (0,7); inconsistent; independent. b. (0, −1); inconsistent; independent. c. No solution; consistent; dependent. d. No solution; inconsistent; independent.
Solve this system of equations by graphing and classifying the system as consistent or inconsistent and the equations as dependent or independent. y=x+3 Equation 1 y = −x + 5 Equation 2 11 10 9 8 7 6 5 4 3 2 1 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 −2
1
2
3
4
5
6
7
8
Solving Systems of Linear Equations Algebraically Notice that the co-ordinates of this point are hard to read directly off the graph.
11 10
9 8 7 6 5 4 3 2 1 −8
−7
−6
−5
−4
−3
−2
−1
0 −1
1
2
3
4
5
6
7
8
−2
In a lot of cases, using algebraic methods to solve linear equations provides more accurate answers. Slide: 36
Substitution Method In the substitution method, one variable in one equation is isolated, and its value is substituted into the other equation. Example
y − 2x + 6 = 0 3y − 5x + 12 = 0
From Equation 1:
Equation 1 Equation 2
y = 2x − 6
Substituting into Equation 2:
3(2x − 6) − 5x + 12 = 0 6x − 18 − 5x + 12 = 0 6x − 5x = 18 − 12 x=6
Substituting x = 6 into Equation 1:
y = 2(6) − 6 y=6
Therefore, the solution is (6, 6). Slide: 37
Solve this system: y + 2x + 4 = 0 −3y − x + 3 = 0 a. (2, −3) b. (−3, 2) 9 46 c. ( 7 , 7 ) d. (−3, −10)
Equation 1 Equation 2
Solve this system: y + 2x + 4 = 0 −3y − x + 3 = 0
Equation 1 Equation 2
Solve this system: 𝟏 𝟗 y=− x– 𝟏 𝟑
𝟒
y= x+ a. (−5, −1) b. (−13, 1) c. (−13, −1) d. (−1, −5)
𝟐 𝟑
𝟒
Equation 1 Equation 2
Solve this system: 𝟏 𝟗 y=− x– 𝟏 𝟑
𝟒
y= x+
𝟐 𝟑
𝟒
Equation 1 Equation 2
Slide: 1
Co-ordinate System Y-axis
Graphs are drawn on what is known as the Cartesian coordinate system. System uses two perpendicular lines. The partitions created on the grid by the lines are known as quadrants. Going counterclockwise from the top right, they are quadrant I, quadrant II, quadrant III, and quadrant IV.
6 5
Quadrant II
3
X: negative Y: positive
X: positive Y: positive
2 1
−9 −8 −7
−6
−5 −4
−3 −2 −1 0 −1
1
2
3
4
5
6
7
−2
Quadrant III X: negative Y: negative
Slide: 2
Quadrant I
4
−3 −4 −5 −6
Quadrant IV X: potitive Y: negative
8
9
X-axis
Co-ordinate System We use ordered pairs to plot points on the graph.
Y-axis 6
(−6,5)
5
Example
(4 , 3) x-value
4
2
y-value
1 −9 −8 −7
Example
(−6 , 5)
(4,3)
3
−6
−5 −4
−3 −2 −1 0 −1
1
2
3
4
5
6
7
8
−2
(−8,−2)
−3 −4
Example
(−8 , −2)
Example
(7 , −4)
Slide: 3
−5 −6
(7,−4)
9
X-axis
Write the co-ordinates of the points shown on the graph. Y-axis
A
6 5
A:
4
E
3 2
B:
C:
D −9 −8 −7
−6
−5 −4
1 −3 −2 −1 0 −1
E:
Slide: 4
1
2
3
−2 −3
D:
C
−4 −5 −6
B
4
5
6
7
8
9
X-axis
Co-ordinate System Y-axis
You can determine the scale of your axes based on the points you need to plot. Example
(–4 , 250)
300 250
(–4,250)
200 150 100 50
−9 −8 −7
Note: The scales of the x and y axes do not need to be the same!
−6
−5 −4
−3 −2 −1 −50 −100 −150 −200 −250 −300
Slide: 5
1
2
3
4
5
6
7
8
9
X-axis
Linear Equations Linear equations consist of only constants and terms containing a single variable raised to the power of 1. Non-linear equations contain at least one term with more than one variable or a variable raised to a power greater than 1.
Linear
Non Linear
3x + 4y = 12
a2 + b2 = c2
a + 2b – 3c = 4d
x2 – 6x = –8
x=5 Slide: 6
a(b – 1) = c
Standard Form of Linear Equations The standard form of a linear equation with 1 variable, x, is as follows:
ax + b = 0 b x =– a
There is a single solution to this equation which can be determined by solving for x.
The standard form of a linear equation with 2 variables, x and y, is as follows:
a x + by + c = 0 Slide: 7
There are infinite pairs of x and y values which will satisfy this equation.
Graphing Linear Equations using a Table of Values Example
Rearrange the following linear equation with 2 variables for y:
6x + 5y = 20 5y = –6x + 20 6 y=– x+4 5
x
y
–5
10
0
4
5
–2
Select values of x that are divisible by 5. In this form, we can easily set up a table of x and y values which satisfy the equation.
Slide: 8
We call x an independent variable and y a dependent variable because the value of y depends on the value we select for x.
Graphing Linear Equations using a Table of Values We can represent the set of all solutions to the linear equation as a line on the Cartesian co-ordinate system.
6x + 5y = 20 Y 12 10 8 6
x
y
–5
10
0
4
5
–2
4 2 X −9 −8 −7
−6
−5 −4
−3 −2 −1 −2 −4 −6 −8 −10 −12
Slide: 9
1
2
3
4
5
6
7
8
9
Step 1: Choose a scale. Step 2: Plot the points. Step 3: Connect the points.
Graph the following linear equation using a table of values.
0 = –3x + 4y + 12
Y 6 5 4 3 2 1 X −9 −8 −7
−6
−5 −4
−3 −2 −1 −1 −2
x
y
−3 −4 −5 −6
Slide: 10
1
2
3
4
5
6
7
8
9
How do you represent the following linear equation on the Cartesian co-ordinate system?
x=4
Y
6
a.
5 4
c.
3 2
d.
1 −9 −8 −7
−6
−5 −4
−3 −2 −1 −1
1
2
3
X 4
5
−2 −3 −4 −5 −6
Slide: 11
b.
6
7
8
9
Intercepts The x-intercept of a linear equation is the point on the graph at which the line passes through the X axis. The y-intercept of a linear equation is the point on the graph at which the line passes through the Y axis. All 2-variable linear equations will have both an x- and y-intercept.
Y
Example
6 5
x-intercept: (4, 0)
4
y-intercept: (0, –3)
2
3
1 X −9 −8 −7
Given a 2-variable linear equation, you can solve for the x-intercept by setting y to 0, and solve for the y-intercept by setting x to 0, and solving the equation. Slide: 12
−6
−5 −4
−3 −2 −1 −1 −2 −3 −4 −5
−6
1
2
3
4
5
6
7
8
9
What are the x- and y- intercepts of the following linear equation?
14x – 4y – 28 = 0 a. x-intercept: (0, –7) y-intercept: (2, 0) b. x-intercept: (–7, 0) y-intercept: (0, 2) c. x-intercept: (0, 2) y-intercept: (–7, 0) d. x-intercept: (2, 0) y-intercept: (0, –7) Slide: 13
What are the x- and y- intercepts of the following linear equation?
Slide: 14
Graphing Linear Equations using Intercepts We can also graph a linear equation by drawing a line that connects the x- and y- intercepts.
Y 12
10 8 6 4 2 X −9 −8 −7
−6
−5 −4
−3 −2 −1 −2
1
2
3
4
5
6
7
8
9
−4 −6 −8
−10 −12
x-intercept: (2, 0) Slide: 15
y-intercept: (0, –7)
Slope-Intercept Form of Linear Equations We have already seen how a 2-variable linear equation in standard form can be rearranged to solve for y. This is called slope-intercept form.
Standard Form:
Slope-Intercept Form:
–3x + 4y + 12 = 0
3 y= x–3 4
Slide: 16
Slope-Intercept Form of Linear Equations We have already seen how a 2-variable linear equation in standard form can be rearranged to solve for y. This is called slope-intercept form.
y = mx + b b is the y-intercept of the line, and m is the slope of the line. The slope refers to the steepness of the line.
Given two points, (x1, y1) and (x2, y2), you can determine the slope of the line using the following formula: Slide: 17
𝑟𝑖𝑠𝑒 𝑦2 − 𝑦1 𝑚= = 𝑟𝑢𝑛 𝑥2 − 𝑥1
Determine the slope of the line containing the following 2 points:
(8, –2), (–2, 6)
Slide: 18
Slope of a Line The sign of m determines the direction of the line.
Positive m:
Negative m:
Y
Y
X
X
y increases
y decreases
As x increases…
Slide: 19
As x increases…
Slope of a Line The greater the absolute value of m, the steeper the line.
m=
𝟏 𝟑
m=3
Y
Y
X
Slide: 20
X
What is the equation of the following line? We have 2 points: (0, –7), (2, 0)
Y 12 10
Determining m:
8
𝑦2 − 𝑦1 0 − (−7) 7 𝑚= = = 𝑥2 − 𝑥1 2−0 2
6 4 2
X −9 −8 −7
Determining b:
−6
−5 −4
−3 −2 −1 −2
rise = 7
The y-intercept is (0, –7).
−4 −6 −8 −10 −12
Slide: 21
run = 2
7 y = x – 7 2
1
2
3
4
5
6
7
8
9
Find the equation of the line that is plotted in the graph shown. Y-axis 11
a. y = 2x
10 9 8
1 7 b. y = 2 x + 2 7 1 c. y = x + 2 2
7 6 5
4 3 2 1 −3 −2 −1 0 −1 −2
d. y = 2x + 6
1
2
3
4
5
6
7
8
9 10 11 12
X-axis
Find the equation of the line that is plotted in the graph shown. Y-axis 11 10 9 8
7 6 5
4 3 2 1 −3 −2 −1 0 −1 −2
1
2
3
4
5
6
7
8
9 10 11 12
X-axis
Parallel Lines Parallel lines have the same slope.
10 9 8
4 5
y= x+6
7
6 5
4 5
y= x+1
4 3 2 1 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 −2
Slide: 24
1
2
3
4
5
6
7
8
These lines are parallel
Perpendicular Lines The product of the slopes of perpendicular lines is –1.
10 9 8
y=
7 6
3 2
4 3 2 1
−2
Slide: 25
+4
y=− x+6
5
0 −8 −7 −6 −5 −4 −3 −2 −1 −1
2 x 3
1
2
3
4
5
6
7
8
These lines are perpendicular because their slopes are negative reciprocals
Write the equation of the line perpendicular to x − 2y = 12 that passes through the point (4,−4).
a. b. c. d.
y = −2x + 4 y = 2x − 12 y = −2x − 4 y = 2x + 12
Write the equation of the line perpendicular to x − 2y = 12 that passes through the point (4,−4).
Systems of Equations Two or more equations analyzed together are called systems of equations.
A system of equations that has one or many solutions is called a consistent system. A system of equations that has no solution is called an inconsistent system.
11 10 9 8 7 6 5 4
If a system of equations has many solutions then the equations are dependent. If a system of equations has one solution then the equations are independent.
Slide: 28
3 2 1 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 −2
1
2
3
4
5
6
7
8
Systems of Equations Lines may be non-parallel and intersecting, parallel and distinct, or parallel and coincident.
Non-Parallel and Intersecting Lines Slopes of lines are different. y-intercepts of lines may or may not be the same.
The system has one solution (the point where the lines intersect). The system is consistent. The equations are independent.
10 9 8 7
One Solution
6 5 4 3 2 1 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 −2
Slide: 29
1
2
3
4
5
6
7
8
Systems of Equations Lines may be non-parallel and intersecting, parallel and distinct, or parallel and coincident.
Parallel and Distinct Lines Slopes of lines are the same. 10
y-intercepts of lines are different.
The system has no solution.
No Solution
9 8 7
The system is inconsistent.
6 5
The equations are independent.
4 3 2 1 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 −2
Slide: 30
1
2
3
4
5
6
7
8
Systems of Equations Lines may be non-parallel and intersecting, parallel and distinct, or parallel and coincident.
Parallel and Coincident Lines Slopes of lines are the same. 10
y-intercepts of lines are the same.
The system has infinitely many solutions.
Many Solutions
9 8 7
The system is consistent.
6 5
The equations are dependent.
4 3 2 1 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 −2
Slide: 31
1
2
3
4
5
6
7
8
Solve this system of equations by graphing and classifying the system as consistent or inconsistent and the equations as dependent or independent. y=x+3 Equation 1 y = −x + 5 Equation 2
a. b. c. d.
(1,4); consistent; independent. (1,4); inconsistent; dependent. (4,1); consistent; independent. (1,4); inconsistent; dependent.
Solve this system of equations by graphing and classifying the system as consistent or inconsistent and the equations as dependent or independent. y=x+3 Equation 1 y = −x + 5 Equation 2 11 10 9 8 7 6 5 4 3 2 1 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 −2
1
2
3
4
5
6
7
8
Solve this system of equations by graphing and classifying the system as consistent or inconsistent and the equations as dependent or independent. y = −3x − 1 Equation 1 y = −3x + 7 Equation 2
a. (0,7); inconsistent; independent. b. (0, −1); inconsistent; independent. c. No solution; consistent; dependent. d. No solution; inconsistent; independent.
Solve this system of equations by graphing and classifying the system as consistent or inconsistent and the equations as dependent or independent. y=x+3 Equation 1 y = −x + 5 Equation 2 11 10 9 8 7 6 5 4 3 2 1 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 −2
1
2
3
4
5
6
7
8
Solving Systems of Linear Equations Algebraically Notice that the co-ordinates of this point are hard to read directly off the graph.
11 10
9 8 7 6 5 4 3 2 1 −8
−7
−6
−5
−4
−3
−2
−1
0 −1
1
2
3
4
5
6
7
8
−2
In a lot of cases, using algebraic methods to solve linear equations provides more accurate answers. Slide: 36
Substitution Method In the substitution method, one variable in one equation is isolated, and its value is substituted into the other equation. Example
y − 2x + 6 = 0 3y − 5x + 12 = 0
From Equation 1:
Equation 1 Equation 2
y = 2x − 6
Substituting into Equation 2:
3(2x − 6) − 5x + 12 = 0 6x − 18 − 5x + 12 = 0 6x − 5x = 18 − 12 x=6
Substituting x = 6 into Equation 1:
y = 2(6) − 6 y=6
Therefore, the solution is (6, 6). Slide: 37
Solve this system: y + 2x + 4 = 0 −3y − x + 3 = 0 a. (2, −3) b. (−3, 2) 9 46 c. ( 7 , 7 ) d. (−3, −10)
Equation 1 Equation 2
Solve this system: y + 2x + 4 = 0 −3y − x + 3 = 0
Equation 1 Equation 2
Solve this system: 𝟏 𝟗 y=− x– 𝟏 𝟑
𝟒
y= x+ a. (−5, −1) b. (−13, 1) c. (−13, −1) d. (−1, −5)
𝟐 𝟑
𝟒
Equation 1 Equation 2
Solve this system: 𝟏 𝟗 y=− x– 𝟏 𝟑
𝟒
y= x+
𝟐 𝟑
𝟒
Equation 1 Equation 2
