Precalculus Examples
[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+b.
y=ax+b
Step 1.1.2
Build a set of equations from the table such that y=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)+b
Step 1.1.3
Calculate the values of a and b.
Step 1.1.3.1
Solve for a in 3=a+b.
Step 1.1.3.1.1
Rewrite the equation as a+b=3.
a+b=3
6=a(2)+b
24=a(8)+b
27=a(9)+b
−6=a(−2)+b
−9=a(−3)+b
Step 1.1.3.1.2
Subtract b from both sides of the equation.
a=3−b
6=a(2)+b
24=a(8)+b
27=a(9)+b
−6=a(−2)+b
−9=a(−3)+b
a=3−b
6=a(2)+b
24=a(8)+b
27=a(9)+b
−6=a(−2)+b
−9=a(−3)+b
Step 1.1.3.2
Replace all occurrences of a with 3−b in each equation.
Step 1.1.3.2.1
Replace all occurrences of a in 6=a(2)+b with 3−b.
6=(3−b)(2)+b
a=3−b
24=a(8)+b
27=a(9)+b
−6=a(−2)+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.
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+b
a=3−b
24=a(8)+b
27=a(9)+b
−6=a(−2)+b
−9=a(−3)+b
Step 1.1.3.2.2.1.1.2
Multiply 3 by 2.
6=6−b⋅2+b
a=3−b
24=a(8)+b
27=a(9)+b
−6=a(−2)+b
−9=a(−3)+b
Step 1.1.3.2.2.1.1.3
Multiply 2 by −1.
6=6−2b+b
a=3−b
24=a(8)+b
27=a(9)+b
−6=a(−2)+b
−9=a(−3)+b
6=6−2b+b
a=3−b
24=a(8)+b
27=a(9)+b
−6=a(−2)+b
−9=a(−3)+b
Step 1.1.3.2.2.1.2
Add −2b and b.
6=6−b
a=3−b
24=a(8)+b
27=a(9)+b
−6=a(−2)+b
−9=a(−3)+b
6=6−b
a=3−b
24=a(8)+b
27=a(9)+b
−6=a(−2)+b
−9=a(−3)+b
6=6−b
a=3−b
24=a(8)+b
27=a(9)+b
−6=a(−2)+b
−9=a(−3)+b
Step 1.1.3.2.3
Replace all occurrences of a in 24=a(8)+b with 3−b.
24=(3−b)(8)+b
6=6−b
a=3−b
27=a(9)+b
−6=a(−2)+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.
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+b
6=6−b
a=3−b
27=a(9)+b
−6=a(−2)+b
−9=a(−3)+b
Step 1.1.3.2.4.1.1.2
Multiply 3 by 8.
24=24−b⋅8+b
6=6−b
a=3−b
27=a(9)+b
−6=a(−2)+b
−9=a(−3)+b
Step 1.1.3.2.4.1.1.3
Multiply 8 by −1.
24=24−8b+b
6=6−b
a=3−b
27=a(9)+b
−6=a(−2)+b
−9=a(−3)+b
24=24−8b+b
6=6−b
a=3−b
27=a(9)+b
−6=a(−2)+b
−9=a(−3)+b
Step 1.1.3.2.4.1.2
Add −8b and b.
24=24−7b
6=6−b
a=3−b
27=a(9)+b
−6=a(−2)+b
−9=a(−3)+b
24=24−7b
6=6−b
a=3−b
27=a(9)+b
−6=a(−2)+b
−9=a(−3)+b
24=24−7b
6=6−b
a=3−b
27=a(9)+b
−6=a(−2)+b
−9=a(−3)+b
Step 1.1.3.2.5
Replace all occurrences of a in 27=a(9)+b with 3−b.
27=(3−b)(9)+b
24=24−7b
6=6−b
a=3−b
−6=a(−2)+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.
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+b
24=24−7b
6=6−b
a=3−b
−6=a(−2)+b
−9=a(−3)+b
Step 1.1.3.2.6.1.1.2
Multiply 3 by 9.
27=27−b⋅9+b
24=24−7b
6=6−b
a=3−b
−6=a(−2)+b
−9=a(−3)+b
Step 1.1.3.2.6.1.1.3
Multiply 9 by −1.
27=27−9b+b
24=24−7b
6=6−b
a=3−b
−6=a(−2)+b
−9=a(−3)+b
27=27−9b+b
24=24−7b
6=6−b
a=3−b
−6=a(−2)+b
−9=a(−3)+b
Step 1.1.3.2.6.1.2
Add −9b and b.
27=27−8b
24=24−7b
6=6−b
a=3−b
−6=a(−2)+b
−9=a(−3)+b
27=27−8b
24=24−7b
6=6−b
a=3−b
−6=a(−2)+b
−9=a(−3)+b
27=27−8b
24=24−7b
6=6−b
a=3−b
−6=a(−2)+b
−9=a(−3)+b
Step 1.1.3.2.7
Replace all occurrences of a in −6=a(−2)+b with 3−b.
−6=(3−b)(−2)+b
27=27−8b
24=24−7b
6=6−b
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.
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
27=27−8b
24=24−7b
6=6−b
a=3−b
−9=a(−3)+b
Step 1.1.3.2.8.1.1.2
Multiply 3 by −2.
−6=−6−b⋅−2+b
27=27−8b
24=24−7b
6=6−b
a=3−b
−9=a(−3)+b
Step 1.1.3.2.8.1.1.3
Multiply −2 by −1.
−6=−6+2b+b
27=27−8b
24=24−7b
6=6−b
a=3−b
−9=a(−3)+b
−6=−6+2b+b
27=27−8b
24=24−7b
6=6−b
a=3−b
−9=a(−3)+b
Step 1.1.3.2.8.1.2
Add 2b and b.
−6=−6+3b
27=27−8b
24=24−7b
6=6−b
a=3−b
−9=a(−3)+b
−6=−6+3b
27=27−8b
24=24−7b
6=6−b
a=3−b
−9=a(−3)+b
−6=−6+3b
27=27−8b
24=24−7b
6=6−b
a=3−b
−9=a(−3)+b
Step 1.1.3.2.9
Replace all occurrences of a in −9=a(−3)+b with 3−b.
−9=(3−b)(−3)+b
−6=−6+3b
27=27−8b
24=24−7b
6=6−b
a=3−b
Step 1.1.3.2.10
Simplify the right side.
Step 1.1.3.2.10.1
Simplify (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
−6=−6+3b
27=27−8b
24=24−7b
6=6−b
a=3−b
Step 1.1.3.2.10.1.1.2
Multiply 3 by −3.
−9=−9−b⋅−3+b
−6=−6+3b
27=27−8b
24=24−7b
6=6−b
a=3−b
Step 1.1.3.2.10.1.1.3
Multiply −3 by −1.
−9=−9+3b+b
−6=−6+3b
27=27−8b
24=24−7b
6=6−b
a=3−b
−9=−9+3b+b
−6=−6+3b
27=27−8b
24=24−7b
6=6−b
a=3−b
Step 1.1.3.2.10.1.2
Add 3b and b.
−9=−9+4b
−6=−6+3b
27=27−8b
24=24−7b
6=6−b
a=3−b
−9=−9+4b
−6=−6+3b
27=27−8b
24=24−7b
6=6−b
a=3−b
−9=−9+4b
−6=−6+3b
27=27−8b
24=24−7b
6=6−b
a=3−b
−9=−9+4b
−6=−6+3b
27=27−8b
24=24−7b
6=6−b
a=3−b
Step 1.1.3.3
Solve for b in −9=−9+4b.
Step 1.1.3.3.1
Rewrite the equation as −9+4b=−9.
−9+4b=−9
−6=−6+3b
27=27−8b
24=24−7b
6=6−b
a=3−b
Step 1.1.3.3.2
Move all terms not containing b to the right side of the equation.
Step 1.1.3.3.2.1
Add 9 to both sides of the equation.
4b=−9+9
−6=−6+3b
27=27−8b
24=24−7b
6=6−b
a=3−b
Step 1.1.3.3.2.2
Add −9 and 9.
4b=0
−6=−6+3b
27=27−8b
24=24−7b
6=6−b
a=3−b
4b=0
−6=−6+3b
27=27−8b
24=24−7b
6=6−b
a=3−b
Step 1.1.3.3.3
Divide each term in 4b=0 by 4 and simplify.
Step 1.1.3.3.3.1
Divide each term in 4b=0 by 4.
4b4=04
−6=−6+3b
27=27−8b
24=24−7b
6=6−b
a=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 4.
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
