Classwork 4-2 Worksheet 1
By Sam Schneider
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Last updated about 1 year ago
16 Questions
1 point
1
Question 1
1.
Examine the following system of linear equations. How many solutions will this system have? That is, how many (x_{1},y_{1}) pairs are there that will work to make all the equations in the system true at the same time? Explain your reasoning.
y=4x+2y=-\frac{1}{2}x-1
Examine the following system of linear equations. How many solutions will this system have? That is, how many (x_{1},y_{1}) pairs are there that will work to make all the equations in the system true at the same time? Explain your reasoning.
y=4x+2
y=-\frac{1}{2}x-1
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Question 2
2.
Assume that the only solution to the system of equations is the ordered pair (\alpha, \beta). That is, each of the equations in the system is true when x=\alpha and y=\beta. Thus, we can write two new equations.
The first equation is \beta=4\alpha+2. Write the second, analogous equation.
Assume that the only solution to the system of equations is the ordered pair (\alpha, \beta). That is, each of the equations in the system is true when x=\alpha and y=\beta. Thus, we can write two new equations.
The first equation is \beta=4\alpha+2. Write the second, analogous equation.
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Question 3
3.
Now combine these two equations to make a new, third equation. We will do this using the "transitive property of equality", which states:
Transitive property of equality:Whenever a=b and b=c, then a=c.
For our purposes, you might prefer to think of this the way Euclid (the famous ancient Greek geometer) phrased it: "If two things are each equal to a third, then they are equal to each other."
Now combine these two equations to make a new, third equation. We will do this using the "transitive property of equality", which states:
Transitive property of equality:
Whenever a=b and b=c, then a=c.
For our purposes, you might prefer to think of this the way Euclid (the famous ancient Greek geometer) phrased it: "If two things are each equal to a third, then they are equal to each other."
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Question 4
4.
Solve your equation from the question above.
Solve your equation from the question above.
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Question 5
5.
If we have a linear equation, y=mx+b, and we know either the y-value or the x-value, we can find the other. Use this fact to continue from the findings above.
If we have a linear equation, y=mx+b, and we know either the y-value or the x-value, we can find the other. Use this fact to continue from the findings above.
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Question 6
6.
What ordered pair, (x,y) is a solution to the system of equations?
What ordered pair, (x,y) is a solution to the system of equations?
SECTION 2 - Practice solving by substitution
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Question 7
7.
Use substitution to find the solution to the following system of linear equations. Then check your answer graphically.
y=4x-3y= -x+2
Use substitution to find the solution to the following system of linear equations. Then check your answer graphically.
y=4x-3
y= -x+2
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Question 8
8.
Use substitution to find the solution to the following system of linear equations. Then check your answer graphically.y= x-3y=\frac{2}{5}x+3
Use substitution to find the solution to the following system of linear equations. Then check your answer graphically.
y= x-3
y=\frac{2}{5}x+3
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Question 9
9.
Use substitution to find the solution to the following system of linear equations. Then check your answer graphically.y=xy= -1x
Use substitution to find the solution to the following system of linear equations. Then check your answer graphically.
y=x
y= -1x
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Question 10
10.
Use substitution to find the solution to the following system of linear equations. Then check your answer graphically.y= -x+14y=2x-22
Use substitution to find the solution to the following system of linear equations. Then check your answer graphically.
y= -x+14
y=2x-22
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Question 11
11.
Use substitution to find the solution to the following system of linear equations. Then check your answer graphically.y= 22x+6y= 44x+6
Use substitution to find the solution to the following system of linear equations. Then check your answer graphically.
y= 22x+6
y= 44x+6
SECTION 3 - Variations & Shortcuts
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Question 12
12.
The following system of equations is written in standard form. As a first step to finding a solution to the system, convert each equation in the system to slope-intercept form.
2x-3y=7-x+5y=2
The following system of equations is written in standard form. As a first step to finding a solution to the system, convert each equation in the system to slope-intercept form.
2x-3y=7
-x+5y=2
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Question 13
13.
Use the slope-intercept form equations to find the solution to the system.
Use the slope-intercept form equations to find the solution to the system.
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Question 14
14.
The following system of equations includes linear equations in different forms. Rewrite ONLY THE FIRST of the equations in slope-intercept form.
y-2=3(x+1)3x+y=9
The following system of equations includes linear equations in different forms. Rewrite ONLY THE FIRST of the equations in slope-intercept form.
y-2=3(x+1)
3x+y=9
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Question 15
15.
Now that you have an equation which provides an equivalent expression for y, substitute that expression for y in the second equation from above. Write your resulting equation here.
Now that you have an equation which provides an equivalent expression for y, substitute that expression for y in the second equation from above. Write your resulting equation here.
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Question 16
16.
What is the solution to the system of equations?
What is the solution to the system of equations?