Solve each equation. Remember to check for extraneous solutions.
1) sqrt(81x) – 5 = 4
2) sqrt(2a – 17) = sqrt(10 – a)
3) – 4 = – 9 + sqrt(3n + 7)
4) 6 = 1 + sqrt(5x)
5) 49 = 7sqrt(- 1 – 10n)
6) sqrt(19a – 2) + 5 = 11
7) sqrt(15 – r) = sqrt(2r – 15)
8) sqrt(2x + 7) = sqrt(3x + 6)
9) sqrt(3a – 13) = sqrt(2a – 8)
10) 3 = sqrt(3v)
The Correct Answer and Explanation is :
Let’s solve each equation step by step and then check for any extraneous solutions that might arise.
1) ( \sqrt{81x} – 5 = 4 )
Step 1: Add 5 to both sides:
[
\sqrt{81x} = 9
]
Step 2: Square both sides to eliminate the square root:
[
81x = 81
]
Step 3: Divide by 81:
[
x = 1
]
Check: Substitute ( x = 1 ) back into the original equation:
[
\sqrt{81(1)} – 5 = 4 \quad \Rightarrow \quad 9 – 5 = 4
]
Since the original equation holds true, the solution is correct: ( x = 1 ).
2) ( \sqrt{2a – 17} = \sqrt{10 – a} )
Step 1: Square both sides:
[
2a – 17 = 10 – a
]
Step 2: Add ( a ) to both sides:
[
3a – 17 = 10
]
Step 3: Add 17 to both sides:
[
3a = 27
]
Step 4: Divide by 3:
[
a = 9
]
Check: Substitute ( a = 9 ) back into the original equation:
[
\sqrt{2(9) – 17} = \sqrt{10 – 9} \quad \Rightarrow \quad \sqrt{18 – 17} = \sqrt{1} \quad \Rightarrow \quad 1 = 1
]
Since this is true, the solution is correct: ( a = 9 ).
3) ( -4 = -9 + \sqrt{3n + 7} )
Step 1: Add 9 to both sides:
[
5 = \sqrt{3n + 7}
]
Step 2: Square both sides:
[
25 = 3n + 7
]
Step 3: Subtract 7 from both sides:
[
18 = 3n
]
Step 4: Divide by 3:
[
n = 6
]
Check: Substitute ( n = 6 ) back into the original equation:
[
-4 = -9 + \sqrt{3(6) + 7} \quad \Rightarrow \quad -4 = -9 + \sqrt{18 + 7} \quad \Rightarrow \quad -4 = -9 + \sqrt{25} \quad \Rightarrow \quad -4 = -9 + 5
]
Since this is true, the solution is correct: ( n = 6 ).
4) ( 6 = 1 + \sqrt{5x} )
Step 1: Subtract 1 from both sides:
[
5 = \sqrt{5x}
]
Step 2: Square both sides:
[
25 = 5x
]
Step 3: Divide by 5:
[
x = 5
]
Check: Substitute ( x = 5 ) back into the original equation:
[
6 = 1 + \sqrt{5(5)} \quad \Rightarrow \quad 6 = 1 + \sqrt{25} \quad \Rightarrow \quad 6 = 1 + 5
]
Since this is true, the solution is correct: ( x = 5 ).
5) ( 49 = 7\sqrt{-1 – 10n} )
Step 1: Divide by 7:
[
7 = \sqrt{-1 – 10n}
]
Step 2: Square both sides:
[
49 = -1 – 10n
]
Step 3: Add 1 to both sides:
[
50 = -10n
]
Step 4: Divide by -10:
[
n = -5
]
Check: Substitute ( n = -5 ) back into the original equation:
[
49 = 7\sqrt{-1 – 10(-5)} \quad \Rightarrow \quad 49 = 7\sqrt{-1 + 50} \quad \Rightarrow \quad 49 = 7\sqrt{49} \quad \Rightarrow \quad 49 = 7 \times 7
]
Since this is true, the solution is correct: ( n = -5 ).
6) ( \sqrt{19a – 2} + 5 = 11 )
Step 1: Subtract 5 from both sides:
[
\sqrt{19a – 2} = 6
]
Step 2: Square both sides:
[
19a – 2 = 36
]
Step 3: Add 2 to both sides:
[
19a = 38
]
Step 4: Divide by 19:
[
a = 2
]
Check: Substitute ( a = 2 ) back into the original equation:
[
\sqrt{19(2) – 2} + 5 = 11 \quad \Rightarrow \quad \sqrt{38 – 2} + 5 = 11 \quad \Rightarrow \quad \sqrt{36} + 5 = 11 \quad \Rightarrow \quad 6 + 5 = 11
]
Since this is true, the solution is correct: ( a = 2 ).
7) ( \sqrt{15 – r} = \sqrt{2r – 15} )
Step 1: Square both sides:
[
15 – r = 2r – 15
]
Step 2: Add ( r ) to both sides:
[
15 = 3r – 15
]
Step 3: Add 15 to both sides:
[
30 = 3r
]
Step 4: Divide by 3:
[
r = 10
]
Check: Substitute ( r = 10 ) back into the original equation:
[
\sqrt{15 – 10} = \sqrt{2(10) – 15} \quad \Rightarrow \quad \sqrt{5} = \sqrt{20 – 15} \quad \Rightarrow \quad \sqrt{5} = \sqrt{5}
]
Since this is true, the solution is correct: ( r = 10 ).
8) ( \sqrt{2x + 7} = \sqrt{3x + 6} )
Step 1: Square both sides:
[
2x + 7 = 3x + 6
]
Step 2: Subtract ( 2x ) from both sides:
[
7 = x + 6
]
Step 3: Subtract 6 from both sides:
[
x = 1
]
Check: Substitute ( x = 1 ) back into the original equation:
[
\sqrt{2(1) + 7} = \sqrt{3(1) + 6} \quad \Rightarrow \quad \sqrt{2 + 7} = \sqrt{3 + 6} \quad \Rightarrow \quad \sqrt{9} = \sqrt{9} \quad \Rightarrow \quad 3 = 3
]
Since this is true, the solution is correct: ( x = 1 ).
9) ( \sqrt{3a – 13} = \sqrt{2a – 8} )
Step 1: Square both sides:
[
3a – 13 = 2a – 8
]
Step 2: Subtract ( 2a ) from both sides:
[
a – 13 = -8
]
Step 3: Add 13 to both sides:
[
a = 5
]
Check: Substitute ( a = 5 ) back into the original equation:
[
\sqrt{3(5) – 13} = \sqrt{2(5) – 8} \quad \Rightarrow \quad \sqrt{15 – 13} = \sqrt{10 – 8} \quad \Rightarrow \quad \sqrt{2} = \sqrt{2}
]
Since this is true, the solution is correct: ( a = 5 ).
10) ( 3 = \sqrt{3v} )
Step 1: Square both sides:
[
9 = 3v
]
Step 2: Divide by 3:
[
v = 3
]
Check: Substitute ( v = 3 ) back into the original equation:
[
3 = \sqrt{3(3)} \quad \Rightarrow \quad 3 = \sqrt{9} \quad \Rightarrow \quad 3 = 3
]
Since this is true, the solution is correct: ( v = 3 ).
Summary of solutions:
- ( x = 1 )
- ( a = 9 )
- ( n = 6 )
- ( x = 5 )
- ( n = -5 )
- ( a = 2 )
- ( r = 10 )
- ( x = 1 )
- ( a = 5 )
- ( v = 3 )
All solutions are valid with no extraneous solutions!