Precalculus Examples
[12789-2-3111]=[x6z2427-6-9333][12789−2−3111]=[x6z2427−6−9333]
Step 1
Step 1.1
Check if the function rule is linear.
Step 1.1.1
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 1.1.2
Build a set of equations from the table such that y=ax+by=ax+b.
6=a(2)+b24=a(8)+b27=a(9)+b-6=a(-2)+b-9=a(-3)+b3=a(1)+b3=a(1)+b3=a(1)+b6=a(2)+b24=a(8)+b27=a(9)+b−6=a(−2)+b−9=a(−3)+b3=a(1)+b3=a(1)+b3=a(1)+b
Step 1.1.3
Calculate the values of aa and bb.
Step 1.1.3.1
Solve for aa in 3=a+b3=a+b.
Step 1.1.3.1.1
Rewrite the equation as a+b=3a+b=3.
a+b=3a+b=3
6=a(2)+b6=a(2)+b
24=a(8)+b24=a(8)+b
27=a(9)+b27=a(9)+b
-6=a(-2)+b−6=a(−2)+b
-9=a(-3)+b−9=a(−3)+b
Step 1.1.3.1.2
Subtract bb from both sides of the equation.
a=3-ba=3−b
6=a(2)+b6=a(2)+b
24=a(8)+b24=a(8)+b
27=a(9)+b27=a(9)+b
-6=a(-2)+b−6=a(−2)+b
-9=a(-3)+b−9=a(−3)+b
a=3-ba=3−b
6=a(2)+b6=a(2)+b
24=a(8)+b24=a(8)+b
27=a(9)+b27=a(9)+b
-6=a(-2)+b−6=a(−2)+b
-9=a(-3)+b−9=a(−3)+b
Step 1.1.3.2
Replace all occurrences of aa with 3-b3−b in each equation.
Step 1.1.3.2.1
Replace all occurrences of aa in 6=a(2)+b6=a(2)+b with 3-b3−b.
6=(3-b)(2)+b6=(3−b)(2)+b
a=3-ba=3−b
24=a(8)+b24=a(8)+b
27=a(9)+b27=a(9)+b
-6=a(-2)+b−6=a(−2)+b
-9=a(-3)+b−9=a(−3)+b
Step 1.1.3.2.2
Simplify the right side.
Step 1.1.3.2.2.1
Simplify (3-b)(2)+b(3−b)(2)+b.
Step 1.1.3.2.2.1.1
Simplify each term.
Step 1.1.3.2.2.1.1.1
Apply the distributive property.
6=3⋅2-b⋅2+b6=3⋅2−b⋅2+b
a=3-ba=3−b
24=a(8)+b24=a(8)+b
27=a(9)+b27=a(9)+b
-6=a(-2)+b−6=a(−2)+b
-9=a(-3)+b−9=a(−3)+b
Step 1.1.3.2.2.1.1.2
Multiply 33 by 22.
6=6-b⋅2+b6=6−b⋅2+b
a=3-ba=3−b
24=a(8)+b24=a(8)+b
27=a(9)+b27=a(9)+b
-6=a(-2)+b−6=a(−2)+b
-9=a(-3)+b−9=a(−3)+b
Step 1.1.3.2.2.1.1.3
Multiply 22 by -1−1.
6=6-2b+b6=6−2b+b
a=3-ba=3−b
24=a(8)+b24=a(8)+b
27=a(9)+b27=a(9)+b
-6=a(-2)+b−6=a(−2)+b
-9=a(-3)+b−9=a(−3)+b
6=6-2b+b6=6−2b+b
a=3-ba=3−b
24=a(8)+b24=a(8)+b
27=a(9)+b27=a(9)+b
-6=a(-2)+b−6=a(−2)+b
-9=a(-3)+b−9=a(−3)+b
Step 1.1.3.2.2.1.2
Add -2b−2b and bb.
6=6-b6=6−b
a=3-ba=3−b
24=a(8)+b24=a(8)+b
27=a(9)+b27=a(9)+b
-6=a(-2)+b−6=a(−2)+b
-9=a(-3)+b−9=a(−3)+b
6=6-b6=6−b
a=3-ba=3−b
24=a(8)+b24=a(8)+b
27=a(9)+b27=a(9)+b
-6=a(-2)+b−6=a(−2)+b
-9=a(-3)+b−9=a(−3)+b
6=6-b6=6−b
a=3-ba=3−b
24=a(8)+b24=a(8)+b
27=a(9)+b27=a(9)+b
-6=a(-2)+b−6=a(−2)+b
-9=a(-3)+b−9=a(−3)+b
Step 1.1.3.2.3
Replace all occurrences of aa in 24=a(8)+b24=a(8)+b with 3-b3−b.
24=(3-b)(8)+b24=(3−b)(8)+b
6=6-b6=6−b
a=3-ba=3−b
27=a(9)+b27=a(9)+b
-6=a(-2)+b−6=a(−2)+b
-9=a(-3)+b−9=a(−3)+b
Step 1.1.3.2.4
Simplify the right side.
Step 1.1.3.2.4.1
Simplify (3-b)(8)+b(3−b)(8)+b.
Step 1.1.3.2.4.1.1
Simplify each term.
Step 1.1.3.2.4.1.1.1
Apply the distributive property.
24=3⋅8-b⋅8+b24=3⋅8−b⋅8+b
6=6-b6=6−b
a=3-ba=3−b
27=a(9)+b27=a(9)+b
-6=a(-2)+b−6=a(−2)+b
-9=a(-3)+b−9=a(−3)+b
Step 1.1.3.2.4.1.1.2
Multiply 33 by 88.
24=24-b⋅8+b24=24−b⋅8+b
6=6-b6=6−b
a=3-ba=3−b
27=a(9)+b27=a(9)+b
-6=a(-2)+b−6=a(−2)+b
-9=a(-3)+b−9=a(−3)+b
Step 1.1.3.2.4.1.1.3
Multiply 88 by -1−1.
24=24-8b+b24=24−8b+b
6=6-b6=6−b
a=3-ba=3−b
27=a(9)+b27=a(9)+b
-6=a(-2)+b−6=a(−2)+b
-9=a(-3)+b−9=a(−3)+b
24=24-8b+b24=24−8b+b
6=6-b6=6−b
a=3-ba=3−b
27=a(9)+b27=a(9)+b
-6=a(-2)+b−6=a(−2)+b
-9=a(-3)+b−9=a(−3)+b
Step 1.1.3.2.4.1.2
Add -8b−8b and bb.
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
27=a(9)+b27=a(9)+b
-6=a(-2)+b−6=a(−2)+b
-9=a(-3)+b−9=a(−3)+b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
27=a(9)+b27=a(9)+b
-6=a(-2)+b−6=a(−2)+b
-9=a(-3)+b−9=a(−3)+b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
27=a(9)+b27=a(9)+b
-6=a(-2)+b−6=a(−2)+b
-9=a(-3)+b−9=a(−3)+b
Step 1.1.3.2.5
Replace all occurrences of aa in 27=a(9)+b27=a(9)+b with 3-b3−b.
27=(3-b)(9)+b27=(3−b)(9)+b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
-6=a(-2)+b−6=a(−2)+b
-9=a(-3)+b−9=a(−3)+b
Step 1.1.3.2.6
Simplify the right side.
Step 1.1.3.2.6.1
Simplify (3-b)(9)+b(3−b)(9)+b.
Step 1.1.3.2.6.1.1
Simplify each term.
Step 1.1.3.2.6.1.1.1
Apply the distributive property.
27=3⋅9-b⋅9+b27=3⋅9−b⋅9+b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
-6=a(-2)+b−6=a(−2)+b
-9=a(-3)+b−9=a(−3)+b
Step 1.1.3.2.6.1.1.2
Multiply 33 by 99.
27=27-b⋅9+b27=27−b⋅9+b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
-6=a(-2)+b−6=a(−2)+b
-9=a(-3)+b−9=a(−3)+b
Step 1.1.3.2.6.1.1.3
Multiply 99 by -1−1.
27=27-9b+b27=27−9b+b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
-6=a(-2)+b−6=a(−2)+b
-9=a(-3)+b−9=a(−3)+b
27=27-9b+b27=27−9b+b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
-6=a(-2)+b−6=a(−2)+b
-9=a(-3)+b−9=a(−3)+b
Step 1.1.3.2.6.1.2
Add -9b−9b and bb.
27=27-8b27=27−8b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
-6=a(-2)+b−6=a(−2)+b
-9=a(-3)+b−9=a(−3)+b
27=27-8b27=27−8b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
-6=a(-2)+b−6=a(−2)+b
-9=a(-3)+b−9=a(−3)+b
27=27-8b27=27−8b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
-6=a(-2)+b−6=a(−2)+b
-9=a(-3)+b−9=a(−3)+b
Step 1.1.3.2.7
Replace all occurrences of aa in -6=a(-2)+b−6=a(−2)+b with 3-b3−b.
-6=(3-b)(-2)+b−6=(3−b)(−2)+b
27=27-8b27=27−8b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
-9=a(-3)+b−9=a(−3)+b
Step 1.1.3.2.8
Simplify the right side.
Step 1.1.3.2.8.1
Simplify (3-b)(-2)+b(3−b)(−2)+b.
Step 1.1.3.2.8.1.1
Simplify each term.
Step 1.1.3.2.8.1.1.1
Apply the distributive property.
-6=3⋅-2-b⋅-2+b−6=3⋅−2−b⋅−2+b
27=27-8b27=27−8b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
-9=a(-3)+b−9=a(−3)+b
Step 1.1.3.2.8.1.1.2
Multiply 33 by -2−2.
-6=-6-b⋅-2+b−6=−6−b⋅−2+b
27=27-8b27=27−8b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
-9=a(-3)+b−9=a(−3)+b
Step 1.1.3.2.8.1.1.3
Multiply -2−2 by -1−1.
-6=-6+2b+b−6=−6+2b+b
27=27-8b27=27−8b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
-9=a(-3)+b−9=a(−3)+b
-6=-6+2b+b−6=−6+2b+b
27=27-8b27=27−8b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
-9=a(-3)+b−9=a(−3)+b
Step 1.1.3.2.8.1.2
Add 2b2b and bb.
-6=-6+3b−6=−6+3b
27=27-8b27=27−8b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
-9=a(-3)+b−9=a(−3)+b
-6=-6+3b−6=−6+3b
27=27-8b27=27−8b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
-9=a(-3)+b−9=a(−3)+b
-6=-6+3b−6=−6+3b
27=27-8b27=27−8b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
-9=a(-3)+b−9=a(−3)+b
Step 1.1.3.2.9
Replace all occurrences of aa in -9=a(-3)+b−9=a(−3)+b with 3-b3−b.
-9=(3-b)(-3)+b−9=(3−b)(−3)+b
-6=-6+3b−6=−6+3b
27=27-8b27=27−8b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
Step 1.1.3.2.10
Simplify the right side.
Step 1.1.3.2.10.1
Simplify (3-b)(-3)+b(3−b)(−3)+b.
Step 1.1.3.2.10.1.1
Simplify each term.
Step 1.1.3.2.10.1.1.1
Apply the distributive property.
-9=3⋅-3-b⋅-3+b−9=3⋅−3−b⋅−3+b
-6=-6+3b−6=−6+3b
27=27-8b27=27−8b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
Step 1.1.3.2.10.1.1.2
Multiply 33 by -3−3.
-9=-9-b⋅-3+b−9=−9−b⋅−3+b
-6=-6+3b−6=−6+3b
27=27-8b27=27−8b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
Step 1.1.3.2.10.1.1.3
Multiply -3−3 by -1−1.
-9=-9+3b+b−9=−9+3b+b
-6=-6+3b−6=−6+3b
27=27-8b27=27−8b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
-9=-9+3b+b−9=−9+3b+b
-6=-6+3b−6=−6+3b
27=27-8b27=27−8b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
Step 1.1.3.2.10.1.2
Add 3b3b and bb.
-9=-9+4b−9=−9+4b
-6=-6+3b−6=−6+3b
27=27-8b27=27−8b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
-9=-9+4b−9=−9+4b
-6=-6+3b−6=−6+3b
27=27-8b27=27−8b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
-9=-9+4b−9=−9+4b
-6=-6+3b−6=−6+3b
27=27-8b27=27−8b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
-9=-9+4b−9=−9+4b
-6=-6+3b−6=−6+3b
27=27-8b27=27−8b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
Step 1.1.3.3
Solve for bb in -9=-9+4b−9=−9+4b.
Step 1.1.3.3.1
Rewrite the equation as -9+4b=-9−9+4b=−9.
-9+4b=-9−9+4b=−9
-6=-6+3b−6=−6+3b
27=27-8b27=27−8b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
Step 1.1.3.3.2
Move all terms not containing bb to the right side of the equation.
Step 1.1.3.3.2.1
Add 99 to both sides of the equation.
4b=-9+94b=−9+9
-6=-6+3b−6=−6+3b
27=27-8b27=27−8b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
Step 1.1.3.3.2.2
Add -9−9 and 99.
4b=04b=0
-6=-6+3b−6=−6+3b
27=27-8b27=27−8b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
4b=04b=0
-6=-6+3b−6=−6+3b
27=27-8b27=27−8b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
Step 1.1.3.3.3
Divide each term in 4b=04b=0 by 44 and simplify.
Step 1.1.3.3.3.1
Divide each term in 4b=04b=0 by 44.
4b4=044b4=04
-6=-6+3b−6=−6+3b
27=27-8b27=27−8b
24=24-7b24=24−7b
6=6-b6=6−b
a=3-ba=3−b
Step 1.1.3.3.3.2
Simplify the left side.
Step 1.1.3.3.3.2.1
Cancel the common factor of 44.
Step 1.1.3.3.3.2.1.1
Cancel the common factor.
4b4=04
-6=-6+3b
27=27-8b
24=24-7b
6=6-b
a=3-b
Step 1.1.3.3.3.2.1.2
Divide b by 1.
b=04
-6=-6+3b
27=27-8b
24=24-7b
6=6-b
a=3-b
b=04
-6=-6+3b
27=27-8b
24=24-7b
6=6-b
a=3-b
b=04
-6=-6+3b
27=27-8b
24=24-7b
6=6-b
a=3-b
Step 1.1.3.3.3.3
Simplify the right side.
Step 1.1.3.3.3.3.1
Divide 0 by 4.
b=0
-6=-6+3b
27=27-8b
24=24-7b
6=6-b
a=3-b
b=0
-6=-6+3b
27=27-8b
24=24-7b
6=6-b
a=3-b
b=0
-6=-6+3b
27=27-8b
24=24-7b
6=6-b
a=3-b
b=0
-6=-6+3b
27=27-8b
24=24-7b
6=6-b
a=3-b
Step 1.1.3.4
Replace all occurrences of b with 0 in each equation.
Step 1.1.3.4.1
Replace all occurrences of b in -6=-6+3b with 0.
-6=-6+3(0)
b=0
27=27-8b
24=24-7b
6=6-b
a=3-b
Step 1.1.3.4.2
Simplify the right side.
Step 1.1.3.4.2.1
Simplify -6+3(0).
Step 1.1.3.4.2.1.1
Multiply 3 by 0.
-6=-6+0
b=0
27=27-8b
24=24-7b
6=6-b
a=3-b
Step 1.1.3.4.2.1.2
Add -6 and 0.
-6=-6
b=0
27=27-8b
24=24-7b
6=6-b
a=3-b
-6=-6
b=0
27=27-8b
24=24-7b
6=6-b
a=3-b
-6=-6
b=0
27=27-8b
24=24-7b
6=6-b
a=3-b
Step 1.1.3.4.3
Replace all occurrences of b in 27=27-8b with 0.
27=27-8⋅0
-6=-6
b=0
24=24-7b
6=6-b
a=3-b
Step 1.1.3.4.4
Simplify the right side.
Step 1.1.3.4.4.1
Simplify 27-8⋅0.
Step 1.1.3.4.4.1.1
Multiply -8 by 0.
27=27+0
-6=-6
b=0
24=24-7b
6=6-b
a=3-b
Step 1.1.3.4.4.1.2
Add 27 and 0.
27=27
-6=-6
b=0
24=24-7b
6=6-b
a=3-b
27=27
-6=-6
b=0
24=24-7b
6=6-b
a=3-b
27=27
-6=-6
b=0
24=24-7b
6=6-b
a=3-b
Step 1.1.3.4.5
Replace all occurrences of b in 24=24-7b with 0.
24=24-7⋅0
27=27
-6=-6
b=0
6=6-b
a=3-b
Step 1.1.3.4.6
Simplify the right side.
Step 1.1.3.4.6.1
Simplify 24-7⋅0.
Step 1.1.3.4.6.1.1
Multiply -7 by 0.
24=24+0
27=27
-6=-6
b=0
6=6-b
a=3-b
Step 1.1.3.4.6.1.2
Add 24 and 0.
24=24
27=27
-6=-6
b=0
6=6-b
a=3-b
24=24
27=27
-6=-6
b=0
6=6-b
a=3-b
24=24
27=27
-6=-6
b=0
6=6-b
a=3-b
Step 1.1.3.4.7
Replace all occurrences of b in 6=6-b with 0.
6=6-(0)
24=24
27=27
-6=-6
b=0
a=3-b
Step 1.1.3.4.8
Simplify the right side.
Step 1.1.3.4.8.1
Subtract 0 from 6.
6=6
24=24
27=27
-6=-6
b=0
a=3-b
6=6
24=24
27=27
-6=-6
b=0
a=3-b
Step 1.1.3.4.9
Replace all occurrences of b in a=3-b with 0.
a=3-(0)
6=6
24=24
27=27
-6=-6
b=0
Step 1.1.3.4.10
Simplify the right side.
Step 1.1.3.4.10.1
Subtract 0 from 3.
a=3
6=6
24=24
27=27
-6=-6
b=0
a=3
6=6
24=24
27=27
-6=-6
b=0
a=3
6=6
24=24
27=27
-6=-6
b=0
Step 1.1.3.5
Remove any equations from the system that are always true.
a=3
b=0
Step 1.1.3.6
List all of the solutions.
a=3,b=0
a=3,b=0
Step 1.1.4
Calculate the value of y using each x value in the relation and compare this value to the given y value in the relation.
Step 1.1.4.1
Calculate the value of y when a=3, b=0, and x=2.
Step 1.1.4.1.1
Multiply 3 by 2.
y=6+0
Step 1.1.4.1.2
Add 6 and 0.
y=6
y=6
Step 1.1.4.2
If the table has a linear function rule, y=y for the corresponding x value, x=2. This check passes since y=6 and y=6.
6=6
Step 1.1.4.3
Calculate the value of y when a=3, b=0, and x=8.
Step 1.1.4.3.1
Multiply 3 by 8.
y=24+0
Step 1.1.4.3.2
Add 24 and 0.
y=24
y=24
Step 1.1.4.4
If the table has a linear function rule, y=y for the corresponding x value, x=8. This check passes since y=24 and y=24.
24=24
Step 1.1.4.5
Calculate the value of y when a=3, b=0, and x=9.
Step 1.1.4.5.1
Multiply 3 by 9.
y=27+0
Step 1.1.4.5.2
Add 27 and 0.
y=27
y=27
Step 1.1.4.6
If the table has a linear function rule, y=y for the corresponding x value, x=9. This check passes since y=27 and y=27.
27=27
Step 1.1.4.7
Calculate the value of y when a=3, b=0, and x=-2.
Step 1.1.4.7.1
Multiply 3 by -2.
y=-6+0
Step 1.1.4.7.2
Add -6 and 0.
y=-6
y=-6
Step 1.1.4.8
If the table has a linear function rule, y=y for the corresponding x value, x=-2. This check passes since y=-6 and y=-6.
-6=-6
Step 1.1.4.9
Calculate the value of y when a=3, b=0, and x=-3.
Step 1.1.4.9.1
Multiply 3 by -3.
y=-9+0
Step 1.1.4.9.2
Add -9 and 0.
y=-9
y=-9
Step 1.1.4.10
If the table has a linear function rule, y=y for the corresponding x value, x=-3. This check passes since y=-9 and y=-9.
-9=-9
Step 1.1.4.11
Calculate the value of y when a=3, b=0, and x=1.
Step 1.1.4.11.1
Multiply 3 by 1.
y=3+0
Step 1.1.4.11.2
Add 3 and 0.
y=3
y=3
Step 1.1.4.12
If the table has a linear function rule, y=y for the corresponding x value, x=1. This check passes since y=3 and y=3.
3=3
Step 1.1.4.13
Calculate the value of y when a=3, b=0, and x=1.
Step 1.1.4.13.1
Multiply 3 by 1.
y=3+0
Step 1.1.4.13.2
Add 3 and 0.
y=3
y=3
Step 1.1.4.14
If the table has a linear function rule, y=y for the corresponding x value, x=1. This check passes since y=3 and y=3.
3=3
Step 1.1.4.15
Calculate the value of y when a=3, b=0, and x=1.
Step 1.1.4.15.1
Multiply 3 by 1.
y=3+0
Step 1.1.4.15.2
Add 3 and 0.
y=3
y=3
Step 1.1.4.16
If the table has a linear function rule, y=y for the corresponding x value, x=1. This check passes since y=3 and y=3.
3=3
Step 1.1.4.17
Since y=y for the corresponding x values, the function is linear.
The function is linear
The function is linear
The function is linear
Step 1.2
Since all y=y, the function is linear and follows the form y=3x.
y=3x
y=3x
Step 2
Step 2.1
Use the function rule equation to find x.
x=3(1)
Step 2.2
Simplify.
x=3
x=3
Step 3
Step 3.1
Use the function rule equation to find z.
z=3(7)
Step 3.2
Simplify.
z=21
z=21
Step 4
List all of the solutions.
x=3z=21
