Algebra Examples
xq(x)11223344xq(x)11223344
Step 1
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.
Step 1.3.1
Solve for aa in 1=a+b1=a+b.
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=1−b
2=a(2)+b2=a(2)+b
3=a(3)+b3=a(3)+b
4=a(4)+b4=a(4)+b
a=1-ba=1−b
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-b1−b in each equation.
Step 1.3.2.1
Replace all occurrences of aa in 2=a(2)+b2=a(2)+b with 1-b1−b.
2=(1-b)(2)+b2=(1−b)(2)+b
a=1-ba=1−b
3=a(3)+b3=a(3)+b
4=a(4)+b4=a(4)+b
Step 1.3.2.2
Simplify the right side.
Step 1.3.2.2.1
Simplify (1-b)(2)+b(1−b)(2)+b.
Step 1.3.2.2.1.1
Simplify each term.
Step 1.3.2.2.1.1.1
Apply the distributive property.
2=1⋅2-b⋅2+b2=1⋅2−b⋅2+b
a=1-ba=1−b
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-b⋅2+b2=2−b⋅2+b
a=1-ba=1−b
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 -1−1.
2=2-2b+b2=2−2b+b
a=1-ba=1−b
3=a(3)+b3=a(3)+b
4=a(4)+b4=a(4)+b
2=2-2b+b2=2−2b+b
a=1-ba=1−b
3=a(3)+b3=a(3)+b
4=a(4)+b4=a(4)+b
Step 1.3.2.2.1.2
Add -2b−2b and bb.
2=2-b2=2−b
a=1-ba=1−b
3=a(3)+b3=a(3)+b
4=a(4)+b4=a(4)+b
2=2-b2=2−b
a=1-ba=1−b
3=a(3)+b3=a(3)+b
4=a(4)+b4=a(4)+b
2=2-b2=2−b
a=1-ba=1−b
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-b1−b.
3=(1-b)(3)+b3=(1−b)(3)+b
2=2-b2=2−b
a=1-ba=1−b
4=a(4)+b4=a(4)+b
Step 1.3.2.4
Simplify the right side.
Step 1.3.2.4.1
Simplify (1-b)(3)+b(1−b)(3)+b.
Step 1.3.2.4.1.1
Simplify each term.
Step 1.3.2.4.1.1.1
Apply the distributive property.
3=1⋅3-b⋅3+b3=1⋅3−b⋅3+b
2=2-b2=2−b
a=1-ba=1−b
4=a(4)+b4=a(4)+b
Step 1.3.2.4.1.1.2
Multiply 33 by 11.
3=3-b⋅3+b3=3−b⋅3+b
2=2-b2=2−b
a=1-ba=1−b
4=a(4)+b4=a(4)+b
Step 1.3.2.4.1.1.3
Multiply 33 by -1−1.
3=3-3b+b3=3−3b+b
2=2-b2=2−b
a=1-ba=1−b
4=a(4)+b4=a(4)+b
3=3-3b+b3=3−3b+b
2=2-b2=2−b
a=1-ba=1−b
4=a(4)+b4=a(4)+b
Step 1.3.2.4.1.2
Add -3b−3b and bb.
3=3-2b3=3−2b
2=2-b2=2−b
a=1-ba=1−b
4=a(4)+b4=a(4)+b
3=3-2b3=3−2b
2=2-b2=2−b
a=1-ba=1−b
4=a(4)+b4=a(4)+b
3=3-2b3=3−2b
2=2-b2=2−b
a=1-ba=1−b
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-b1−b.
4=(1-b)(4)+b4=(1−b)(4)+b
3=3-2b3=3−2b
2=2-b2=2−b
a=1-ba=1−b
Step 1.3.2.6
Simplify the right side.
Step 1.3.2.6.1
Simplify (1-b)(4)+b(1−b)(4)+b.
Step 1.3.2.6.1.1
Simplify each term.
Step 1.3.2.6.1.1.1
Apply the distributive property.
4=1⋅4-b⋅4+b4=1⋅4−b⋅4+b
3=3-2b3=3−2b
2=2-b2=2−b
a=1-ba=1−b
Step 1.3.2.6.1.1.2
Multiply 44 by 11.
4=4-b⋅4+b4=4−b⋅4+b
3=3-2b3=3−2b
2=2-b2=2−b
a=1-ba=1−b
Step 1.3.2.6.1.1.3
Multiply 44 by -1−1.
4=4-4b+b4=4−4b+b
3=3-2b3=3−2b
2=2-b2=2−b
a=1-ba=1−b
4=4-4b+b4=4−4b+b
3=3-2b3=3−2b
2=2-b2=2−b
a=1-ba=1−b
Step 1.3.2.6.1.2
Add -4b−4b and bb.
4=4-3b4=4−3b
3=3-2b3=3−2b
2=2-b2=2−b
a=1-ba=1−b
4=4-3b4=4−3b
3=3-2b3=3−2b
2=2-b2=2−b
a=1-ba=1−b
4=4-3b4=4−3b
3=3-2b3=3−2b
2=2-b2=2−b
a=1-ba=1−b
4=4-3b4=4−3b
3=3-2b3=3−2b
2=2-b2=2−b
a=1-ba=1−b
Step 1.3.3
Solve for bb in 4=4-3b4=4−3b.
Step 1.3.3.1
Rewrite the equation as 4-3b=44−3b=4.
4-3b=44−3b=4
3=3-2b3=3−2b
2=2-b2=2−b
a=1-ba=1−b
Step 1.3.3.2
Move all terms not containing bb to the right side of the equation.
Step 1.3.3.2.1
Subtract 44 from both sides of the equation.
-3b=4-4−3b=4−4
3=3-2b3=3−2b
2=2-b2=2−b
a=1-ba=1−b
Step 1.3.3.2.2
Subtract 44 from 44.
-3b=0−3b=0
3=3-2b3=3−2b
2=2-b2=2−b
a=1-ba=1−b
-3b=0−3b=0
3=3-2b3=3−2b
2=2-b2=2−b
a=1-ba=1−b
Step 1.3.3.3
Divide each term in -3b=0−3b=0 by -3−3 and simplify.
Step 1.3.3.3.1
Divide each term in -3b=0−3b=0 by -3−3.
-3b-3=0-3−3b−3=0−3
3=3-2b3=3−2b
2=2-b2=2−b
a=1-ba=1−b
Step 1.3.3.3.2
Simplify the left side.
Step 1.3.3.3.2.1
Cancel the common factor of -3−3.
Step 1.3.3.3.2.1.1
Cancel the common factor.
-3b-3=0-3−3b−3=0−3
3=3-2b3=3−2b
2=2-b2=2−b
a=1-ba=1−b
Step 1.3.3.3.2.1.2
Divide bb by 11.
b=0-3b=0−3
3=3-2b3=3−2b
2=2-b2=2−b
a=1-ba=1−b
b=0-3b=0−3
3=3-2b3=3−2b
2=2-b2=2−b
a=1-ba=1−b
b=0-3b=0−3
3=3-2b3=3−2b
2=2-b2=2−b
a=1-ba=1−b
Step 1.3.3.3.3
Simplify the right side.
Step 1.3.3.3.3.1
Divide 00 by -3−3.
b=0b=0
3=3-2b3=3−2b
2=2-b2=2−b
a=1-ba=1−b
b=0b=0
3=3-2b3=3−2b
2=2-b2=2−b
a=1-ba=1−b
b=0b=0
3=3-2b3=3−2b
2=2-b2=2−b
a=1-ba=1−b
b=0b=0
3=3-2b3=3−2b
2=2-b2=2−b
a=1-ba=1−b
Step 1.3.4
Replace all occurrences of bb with 00 in each equation.
Step 1.3.4.1
Replace all occurrences of bb in 3=3-2b3=3−2b with 00.
3=3-2⋅03=3−2⋅0
b=0b=0
2=2-b2=2−b
a=1-ba=1−b
Step 1.3.4.2
Simplify the right side.
Step 1.3.4.2.1
Simplify 3-2⋅03−2⋅0.
Step 1.3.4.2.1.1
Multiply -2−2 by 00.
3=3+03=3+0
b=0b=0
2=2-b2=2−b
a=1-ba=1−b
Step 1.3.4.2.1.2
Add 33 and 00.
3=33=3
b=0b=0
2=2-b2=2−b
a=1-ba=1−b
3=33=3
b=0b=0
2=2-b2=2−b
a=1-ba=1−b
3=33=3
b=0b=0
2=2-b2=2−b
a=1-ba=1−b
Step 1.3.4.3
Replace all occurrences of bb in 2=2-b2=2−b with 00.
2=2-(0)2=2−(0)
3=33=3
b=0b=0
a=1-ba=1−b
Step 1.3.4.4
Simplify the right side.
Step 1.3.4.4.1
Subtract 00 from 22.
2=22=2
3=33=3
b=0b=0
a=1-ba=1−b
2=22=2
3=33=3
b=0b=0
a=1-ba=1−b
Step 1.3.4.5
Replace all occurrences of bb in a=1-ba=1−b 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.
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.
Step 1.4.1
Calculate the value of yy when a=1a=1, b=0b=0, and x=1x=1.
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.
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.
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.
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