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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.
A system of equations is a set of two or more equations involving the same variables. The solutions to a system of equations are values for these variables that satisfy all the equations simultaneously. A system of equations is usually written as a vertical list with a curly bracket on the left-hand side. 2x-3y=1 3x+y=7 Graphically, solutions to systems of equations are the points where the graphs of the equations intersect. For this reason, these solutions are usually expressed as coordinates.
(I): .LHS /2.=.RHS /2.
(I): Write as a sum of fractions
(I): a* b/c=a/c* 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): a* b/c=a/c* 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.
Recall that the solution to a system is the point where the lines intersect. If a system has no solution, then the lines never intersect. In fact, the lines must be parallel, meaning that they have the same slope and different y-intercepts. Here is an example of one such system. y=3x+2 y=3x-5
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.
In contrast to parallel lines, lines that intersect once must have different slopes. For example, the following system must have exactly one solution because the two lines have different slopes. y=- x+5 y=3x-2
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.
y=3x+1 2y=6x+2First, 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): a* b/c=a/c* 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?
This lesson focused on solving systems of linear equations by graphing. When equations in the system are written in standard or point-slope form, it is unclear how many solutions the system has. However, the situation is different when it comes to slope-intercept form. Slope-Intercept Form y=mx+b If both equations of the system of equations are in slope-intercept form, there is no need to graph them in order to find the number of solutions. It can be found by analyzing the slopes and y-intercepts of the equations.
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 |
One equation in a system is y= 56x-7.
Write a second equation so that the system has one solution, then graph the system.
Write a second equation so that the system has no solutions, then graph the system.
Write a second equation so that the system has infinitely many solutions, then graph the system.
We are asked to write a second equation to form a system with one solution. Let's begin by analyzing the given equation. y=5/6x-7 Recall that lines that have different slopes will always have one point of intersection. This means that if we keep the same y-intercept and change the slope, the only point of intersection will be the shared y-intercept. While we could use a different slope and y-intercept, at minimum, we just need to choose a different slope. Let's use - 2 for our equation. y=- 2x-7 Together the equations form the following system of equations. y= 56x-7 y=- 2x-7 Let's graph the system on a coordinate plane to make sure the lines have only one solution, represented by only one point of intersection.
Keep in mind that the second equation is an example equation, so answers may vary.
We are asked to write a second equation to form a system with no solutions. This means that the two lines must be parallel — in other words, they should have the same slope and cannot share an y-intercept. Let's recall the slope and the y-intercept of the given equation. Slope: & 5/6 [0.5em] y-intercept: & - 7 We need to keep the same slope and choose another y-intercept. Let's use - 4 for our example. y=5/6x-4 This way we get the following system of equations. y= 56x-7 y= 56x-4 Let's now graph both equations.
Keep in mind that the second equation is an example equation, so answers may vary.
We need to write a second equation so that there are infinitely many solutions. This means that the lines must intersect at each and every point on both of the lines. For this to be true, the two lines must be coincidental, or the same line. They should have the same equation or be different forms of the same equation. y=5/6x-7 We can create a different form of this equation by multiplying both sides of the equation by any non-zero number — for example, 6. y * 6=(5/6x-7) * 6 ⇓ 6y=5x-42 This equation and the given one form the following system of equations. y= 56x-7 6y=5x-42 Let's graph this final system of equations to make sure that they are indeed the same line and have infinitely many solutions.
Keep in mind that the second equation is an example equation, so answers may vary.