Algebra Examples

Find the Function Rule
xq(x)11223344xq(x)11223344
Step 1
Check if the function rule is linear.
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Step 1.1
To find if the table follows a function rule, check to see if the values follow the linear form y=ax+by=ax+b.
y=ax+by=ax+b
Step 1.2
Build a set of equations from the table such that q(x)=ax+bq(x)=ax+b.
1=a(1)+b2=a(2)+b3=a(3)+b4=a(4)+b1=a(1)+b2=a(2)+b3=a(3)+b4=a(4)+b
Step 1.3
Calculate the values of aa and bb.
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Step 1.3.1
Solve for aa in 1=a+b1=a+b.
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Step 1.3.1.1
Rewrite the equation as a+b=1a+b=1.
a+b=1a+b=1
2=a(2)+b2=a(2)+b
3=a(3)+b3=a(3)+b
4=a(4)+b4=a(4)+b
Step 1.3.1.2
Subtract bb from both sides of the equation.
a=1-ba=1b
2=a(2)+b2=a(2)+b
3=a(3)+b3=a(3)+b
4=a(4)+b4=a(4)+b
a=1-ba=1b
2=a(2)+b2=a(2)+b
3=a(3)+b3=a(3)+b
4=a(4)+b4=a(4)+b
Step 1.3.2
Replace all occurrences of aa with 1-b1b in each equation.
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Step 1.3.2.1
Replace all occurrences of aa in 2=a(2)+b2=a(2)+b with 1-b1b.
2=(1-b)(2)+b2=(1b)(2)+b
a=1-ba=1b
3=a(3)+b3=a(3)+b
4=a(4)+b4=a(4)+b
Step 1.3.2.2
Simplify the right side.
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Step 1.3.2.2.1
Simplify (1-b)(2)+b(1b)(2)+b.
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Step 1.3.2.2.1.1
Simplify each term.
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Step 1.3.2.2.1.1.1
Apply the distributive property.
2=12-b2+b2=12b2+b
a=1-ba=1b
3=a(3)+b3=a(3)+b
4=a(4)+b4=a(4)+b
Step 1.3.2.2.1.1.2
Multiply 22 by 11.
2=2-b2+b2=2b2+b
a=1-ba=1b
3=a(3)+b3=a(3)+b
4=a(4)+b4=a(4)+b
Step 1.3.2.2.1.1.3
Multiply 22 by -11.
2=2-2b+b2=22b+b
a=1-ba=1b
3=a(3)+b3=a(3)+b
4=a(4)+b4=a(4)+b
2=2-2b+b2=22b+b
a=1-ba=1b
3=a(3)+b3=a(3)+b
4=a(4)+b4=a(4)+b
Step 1.3.2.2.1.2
Add -2b2b and bb.
2=2-b2=2b
a=1-ba=1b
3=a(3)+b3=a(3)+b
4=a(4)+b4=a(4)+b
2=2-b2=2b
a=1-ba=1b
3=a(3)+b3=a(3)+b
4=a(4)+b4=a(4)+b
2=2-b2=2b
a=1-ba=1b
3=a(3)+b3=a(3)+b
4=a(4)+b4=a(4)+b
Step 1.3.2.3
Replace all occurrences of aa in 3=a(3)+b3=a(3)+b with 1-b1b.
3=(1-b)(3)+b3=(1b)(3)+b
2=2-b2=2b
a=1-ba=1b
4=a(4)+b4=a(4)+b
Step 1.3.2.4
Simplify the right side.
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Step 1.3.2.4.1
Simplify (1-b)(3)+b(1b)(3)+b.
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Step 1.3.2.4.1.1
Simplify each term.
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Step 1.3.2.4.1.1.1
Apply the distributive property.
3=13-b3+b3=13b3+b
2=2-b2=2b
a=1-ba=1b
4=a(4)+b4=a(4)+b
Step 1.3.2.4.1.1.2
Multiply 33 by 11.
3=3-b3+b3=3b3+b
2=2-b2=2b
a=1-ba=1b
4=a(4)+b4=a(4)+b
Step 1.3.2.4.1.1.3
Multiply 33 by -11.
3=3-3b+b3=33b+b
2=2-b2=2b
a=1-ba=1b
4=a(4)+b4=a(4)+b
3=3-3b+b3=33b+b
2=2-b2=2b
a=1-ba=1b
4=a(4)+b4=a(4)+b
Step 1.3.2.4.1.2
Add -3b3b and bb.
3=3-2b3=32b
2=2-b2=2b
a=1-ba=1b
4=a(4)+b4=a(4)+b
3=3-2b3=32b
2=2-b2=2b
a=1-ba=1b
4=a(4)+b4=a(4)+b
3=3-2b3=32b
2=2-b2=2b
a=1-ba=1b
4=a(4)+b4=a(4)+b
Step 1.3.2.5
Replace all occurrences of aa in 4=a(4)+b4=a(4)+b with 1-b1b.
4=(1-b)(4)+b4=(1b)(4)+b
3=3-2b3=32b
2=2-b2=2b
a=1-ba=1b
Step 1.3.2.6
Simplify the right side.
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Step 1.3.2.6.1
Simplify (1-b)(4)+b(1b)(4)+b.
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Step 1.3.2.6.1.1
Simplify each term.
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Step 1.3.2.6.1.1.1
Apply the distributive property.
4=14-b4+b4=14b4+b
3=3-2b3=32b
2=2-b2=2b
a=1-ba=1b
Step 1.3.2.6.1.1.2
Multiply 44 by 11.
4=4-b4+b4=4b4+b
3=3-2b3=32b
2=2-b2=2b
a=1-ba=1b
Step 1.3.2.6.1.1.3
Multiply 44 by -11.
4=4-4b+b4=44b+b
3=3-2b3=32b
2=2-b2=2b
a=1-ba=1b
4=4-4b+b4=44b+b
3=3-2b3=32b
2=2-b2=2b
a=1-ba=1b
Step 1.3.2.6.1.2
Add -4b4b and bb.
4=4-3b4=43b
3=3-2b3=32b
2=2-b2=2b
a=1-ba=1b
4=4-3b4=43b
3=3-2b3=32b
2=2-b2=2b
a=1-ba=1b
4=4-3b4=43b
3=3-2b3=32b
2=2-b2=2b
a=1-ba=1b
4=4-3b4=43b
3=3-2b3=32b
2=2-b2=2b
a=1-ba=1b
Step 1.3.3
Solve for bb in 4=4-3b4=43b.
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Step 1.3.3.1
Rewrite the equation as 4-3b=443b=4.
4-3b=443b=4
3=3-2b3=32b
2=2-b2=2b
a=1-ba=1b
Step 1.3.3.2
Move all terms not containing bb to the right side of the equation.
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Step 1.3.3.2.1
Subtract 44 from both sides of the equation.
-3b=4-43b=44
3=3-2b3=32b
2=2-b2=2b
a=1-ba=1b
Step 1.3.3.2.2
Subtract 44 from 44.
-3b=03b=0
3=3-2b3=32b
2=2-b2=2b
a=1-ba=1b
-3b=03b=0
3=3-2b3=32b
2=2-b2=2b
a=1-ba=1b
Step 1.3.3.3
Divide each term in -3b=03b=0 by -33 and simplify.
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Step 1.3.3.3.1
Divide each term in -3b=03b=0 by -33.
-3b-3=0-33b3=03
3=3-2b3=32b
2=2-b2=2b
a=1-ba=1b
Step 1.3.3.3.2
Simplify the left side.
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Step 1.3.3.3.2.1
Cancel the common factor of -33.
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Step 1.3.3.3.2.1.1
Cancel the common factor.
-3b-3=0-33b3=03
3=3-2b3=32b
2=2-b2=2b
a=1-ba=1b
Step 1.3.3.3.2.1.2
Divide bb by 11.
b=0-3b=03
3=3-2b3=32b
2=2-b2=2b
a=1-ba=1b
b=0-3b=03
3=3-2b3=32b
2=2-b2=2b
a=1-ba=1b
b=0-3b=03
3=3-2b3=32b
2=2-b2=2b
a=1-ba=1b
Step 1.3.3.3.3
Simplify the right side.
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Step 1.3.3.3.3.1
Divide 00 by -33.
b=0b=0
3=3-2b3=32b
2=2-b2=2b
a=1-ba=1b
b=0b=0
3=3-2b3=32b
2=2-b2=2b
a=1-ba=1b
b=0b=0
3=3-2b3=32b
2=2-b2=2b
a=1-ba=1b
b=0b=0
3=3-2b3=32b
2=2-b2=2b
a=1-ba=1b
Step 1.3.4
Replace all occurrences of bb with 00 in each equation.
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Step 1.3.4.1
Replace all occurrences of bb in 3=3-2b3=32b with 00.
3=3-203=320
b=0b=0
2=2-b2=2b
a=1-ba=1b
Step 1.3.4.2
Simplify the right side.
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Step 1.3.4.2.1
Simplify 3-20320.
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Step 1.3.4.2.1.1
Multiply -22 by 00.
3=3+03=3+0
b=0b=0
2=2-b2=2b
a=1-ba=1b
Step 1.3.4.2.1.2
Add 33 and 00.
3=33=3
b=0b=0
2=2-b2=2b
a=1-ba=1b
3=33=3
b=0b=0
2=2-b2=2b
a=1-ba=1b
3=33=3
b=0b=0
2=2-b2=2b
a=1-ba=1b
Step 1.3.4.3
Replace all occurrences of bb in 2=2-b2=2b with 00.
2=2-(0)2=2(0)
3=33=3
b=0b=0
a=1-ba=1b
Step 1.3.4.4
Simplify the right side.
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Step 1.3.4.4.1
Subtract 00 from 22.
2=22=2
3=33=3
b=0b=0
a=1-ba=1b
2=22=2
3=33=3
b=0b=0
a=1-ba=1b
Step 1.3.4.5
Replace all occurrences of bb in a=1-ba=1b with 00.
a=1-(0)a=1(0)
2=22=2
3=33=3
b=0b=0
Step 1.3.4.6
Simplify the right side.
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Step 1.3.4.6.1
Subtract 00 from 11.
a=1a=1
2=22=2
3=33=3
b=0b=0
a=1a=1
2=22=2
3=33=3
b=0b=0
a=1a=1
2=22=2
3=33=3
b=0b=0
Step 1.3.5
Remove any equations from the system that are always true.
a=1a=1
b=0b=0
Step 1.3.6
List all of the solutions.
a=1,b=0a=1,b=0
a=1,b=0a=1,b=0
Step 1.4
Calculate the value of yy using each xx value in the relation and compare this value to the given q(x)q(x) value in the relation.
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Step 1.4.1
Calculate the value of yy when a=1a=1, b=0b=0, and x=1x=1.
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Step 1.4.1.1
Multiply 11 by 11.
y=1+0y=1+0
Step 1.4.1.2
Add 11 and 00.
y=1y=1
y=1y=1
Step 1.4.2
If the table has a linear function rule, y=q(x)y=q(x) for the corresponding xx value, x=1x=1. This check passes since y=1y=1 and q(x)=1q(x)=1.
1=11=1
Step 1.4.3
Calculate the value of yy when a=1a=1, b=0b=0, and x=2x=2.
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Step 1.4.3.1
Multiply 22 by 11.
y=2+0y=2+0
Step 1.4.3.2
Add 22 and 00.
y=2y=2
y=2y=2
Step 1.4.4
If the table has a linear function rule, y=q(x)y=q(x) for the corresponding xx value, x=2x=2. This check passes since y=2y=2 and q(x)=2q(x)=2.
2=22=2
Step 1.4.5
Calculate the value of yy when a=1a=1, b=0b=0, and x=3x=3.
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Step 1.4.5.1
Multiply 33 by 11.
y=3+0y=3+0
Step 1.4.5.2
Add 33 and 00.
y=3y=3
y=3y=3
Step 1.4.6
If the table has a linear function rule, y=q(x)y=q(x) for the corresponding xx value, x=3x=3. This check passes since y=3y=3 and q(x)=3q(x)=3.
3=33=3
Step 1.4.7
Calculate the value of yy when a=1a=1, b=0b=0, and x=4x=4.
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Step 1.4.7.1
Multiply 44 by 11.
y=4+0y=4+0
Step 1.4.7.2
Add 44 and 00.
y=4y=4
y=4y=4
Step 1.4.8
If the table has a linear function rule, y=q(x)y=q(x) for the corresponding xx value, x=4x=4. This check passes since y=4y=4 and q(x)=4q(x)=4.
4=44=4
Step 1.4.9
Since y=q(x)y=q(x) for the corresponding xx values, the function is linear.
The function is linear
The function is linear
The function is linear
Step 2
Since all y=q(x)y=q(x), the function is linear and follows the form y=xy=x.
y=xy=x
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