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| 12 Theory slides |
| 10 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
In the graph below, four lines and their corresponding linear equations can be seen on a coordinate plane. Determine which lines intersect at one point, which lines intersect at infinitely many points, and which lines do not intersect at all.
Consider the definition of equations in two variables.
(I): LHS/2=RHS/2
(I): Write as a sum of fractions
(I): ca⋅b=ca⋅b
(I): Put minus sign in front of fraction
(I): Calculate quotient
(I): Identity Property of Multiplication
Now that the equations are both written in slope-intercept form, they can be graphed on the same coordinate plane.
The point where the lines intersect is the solution to the system.
The lines appear to intersect at (1.5,2.5). Therefore, this is the solution to the system — the value of x is 1.5 and the value of y is 2.5.
Mark is throwing a party, so he bought some donuts and lollipops for his friends.
Start by writing both linear equations in slope-intercept form.
(I): LHS−x=RHS−x
(II): LHS−3x=RHS−3x
(II): LHS/2=RHS/2
(II): Write as a sum of fractions
(II): ca⋅b=ca⋅b
(II): Put minus sign in front of fraction
(II): Calculate quotient
Since the number of items cannot be negative, only the first quadrant will be considered for the graph. The y-intercept of the second equation is 26, so the first point on that line is (0,26). The slope is -1.5. To better match the scale of this graph, the second point can be plotted by going 4 steps to the right and 4⋅1.5=6 steps down.
Finally, the point of intersection P can be identified.
The point of intersection of the lines is P(12,8). In the context of the situation, this means that Mark bought x=12 donuts and y=8 lollipops.
Consider the graph of a system of equations consisting of two lines. What is the solution to the system?
When a system of linear equations has two equations and two variables, the system can have zero, one, or infinitely many solutions.
If a system has no solution, its graph might look similar to this one.
If a system of equations has one solution, its graph consists of two lines that intersect exactly once. The point of intersection is the solution to the system.
If a system of equations has infinitely many solutions, the lines intersect at infinitely many points. This means the lines lie on top of each other or coincide with each other.
These lines are said to be coincidental, and since they have the same slope and y-intercept, they are different versions of the same line. Here is one example of a system that has an infinite number of solutions.
First, rewrite the equations into slope-intercept form. Then, graph the equations and find the point of intersection of the lines.
(I): LHS−8q=RHS−8q
(I): LHS/5=RHS/5
(II): Commutative Property of Addition
(II): LHS/12.5=RHS/12.5
(I), (II): Put minus sign in front of fraction
(I), (II): ca⋅b=ca⋅b
(I), (II): Calculate quotient
The lines overlap each other. They intersect at infinitely many points, which means that the system of equations has infinitely many solutions. This indicates that the system of equations was not set up properly. Maybe the same information was written in two different ways, which resulted in two equations that represent the same line.
Begin by rewriting the equations into slope-intercept form. Then, graph both equations on the same coordinate plane using their slopes and y-intercepts.
(I): LHS−9a=RHS−9a
(II): Commutative Property of Addition
(I): LHS/4=RHS/4
(II): LHS/2=RHS/2
(I), (II): Write as a sum
(I), (II): Put minus sign in front of fraction
(I), (II): Calculate quotient
Consider the given system of equations. Does it have zero, one, or infinitely many solutions?
Characteristics | Number of Solutions |
---|---|
Same slope and same y-intercept | Infinitely many solutions |
Same slope and different y-intercepts | No solution |
Different slopes | One solution |
The following graph corresponds to the system of equations next to it.
The solution of the system of linear equations is the point where the graphs of the equations intersect.
From the graph, we can tell that this point is (4,3). Let's check this solution by substituting the point into both equations.
Since both equations hold true after substituting the values of the point, we know that (4,3) is the solution of the system of equations.
The following graph corresponds to the system of equations next to it.
The solution of the system of linear equations is the point where the graphs of the equations intersect.
From the graph, we can tell that this point is (- 5,2). Let's check this solution by substituting the point into both equations.
Since both equations are true after substituting the values of the point, we know that (- 5,2) is indeed the solution of the system of equations.
We can determine the solution to the system by graphing the given equations. This will be the point at which the lines intersect. To do this, we will need the equations to be in slope-intercept form to help us identify the slope m and y-intercept b.
Let's rewrite each of the equations in the system in slope-intercept form, highlighting the m and b values.
Given Equation | Slope-Intercept Form | Slope m | y-intercept b |
---|---|---|---|
5x-5y=20 | y= 1x+( - 4) | 1 | (0, - 4) |
y=- 4 | y= 0x+( - 4) | 0 | (0, - 4) |
We will start graphing the equations by plotting the y-intercepts. From there, we will use the slope of each equation to determine another point on the line and connect the points with a line.
We can see that the lines intersect at exactly one point.
The point of intersection at (0,- 4) is the solution to the system.
We can determine the solution to the system by graphing the given equations. This will be the point at which the lines intersect. Since both equations area already given in slope-intercept form, let's analyze them to identify the slope m and y-intercept b in each equation.
Given Equation | Slope-Intercept Form | Slope m | y-intercept b |
---|---|---|---|
y=2/3x+4 | y= 2/3x+ 4 | 2/3 | (0, 4) |
y=1/3x+1 | y= 1/3x+ 1 | 1/3 | (0, 1) |
Next, let's graph the equations. We will start by plotting the y-intercepts, then use the slope to determine another point that satisfies the equation. Then we can connect the points with a line.
We can see that the lines intersect at exactly one point.
The point of intersection at (- 9,- 2) is the solution to the system.
Solve the system by graphing. Determine whether the system has one solution, no solution, or infinitely many solutions.
We can determine the number of solutions to the system by graphing the given equations. This will be the point at which the lines intersect. To do this, we need the equations to be in slope-intercept form to help us identify the slope colIVm and y-intercept b.
Let's rewrite each of the equations in the system in slope-intercept form, highlighting the m and b values.
Given Equation | Slope-Intercept Form | Slope m | y-intercept b |
---|---|---|---|
y-1.5x=6 | y= 1.5x+ 6 | 1.5 | (0, 6) |
4y=6x+24 | y= 1.5x+ 6 | 1.5 | (0, 6) |
When converted into slope-intercept form, we can see that these two equations are actually the same. This means that there are infinitely many solutions. Let's confirm this by graphing the system.
To graph these equations, we will start by plotting the y-intercepts, then use the slope to determine another point that satisfies the equation. Then we can connect the points with a line.
The equations overlap at every possible point. Since the equations overlap, they also intersect
at every point along the line. Therefore, the system has infinitely many solutions.
We can determine the number of solutions to the system by graphing the given equations. This will be the point at which the lines intersect. To do this, we need the equations to be in slope-intercept form to help us identify the slope m and y-intercept b.
Let's rewrite each of the equations in the system in slope-intercept form, highlighting the m and b values.
Given Equation | Slope-Intercept Form | Slope m | y-intercept b |
---|---|---|---|
6x-2y=4 | y= 3x+( - 2) | 3 | (0, - 2) |
2y=- 1/2x+5/2 | y= -1/4x+ 5/4 | -1/4 | (0, 5/4) |
To graph these equations, we will start by plotting the y-intercepts, then use the slope to determine another point that satisfies the equation. Then we can connect the points with a line.
We can see that the lines intersect at exactly one point.
The point of intersection at (1,1) is the one solution to the system.