Explanation
The following solutions cover equations from Section A, Section B, and Section C as provided. Each is solved step-by-step, and answers are left as simplified fractions or decimals where needed.
Section A
Concepts
Linear Equations, Basic Arithmetic, Fractional Equations, Solving for Unknown
Question 1
Step 1
32x+5=11
Multiply both sides by 3:
(2x+5)=33
Step 2
2x=33−5=28
Step 3
x=228=14
Final Answer
x=14
Question 2
Step 1
28−3x=5
Multiply both sides by 2:
8−3x=10
Step 2
−3x=10−8=2
Step 3
x=−32=−32
Final Answer
x=−32
Question 3
Step 1
65−9x=−2
Multiply both sides by 6:
5−9x=−12
Step 2
−9x=−12−5=−17
Step 3
x=−9−17=917
Final Answer
x=917
Question 4
Step 1
37x+6=−9−12=−21
Multiply both sides by 3:
7x+6=−63
Step 2
7x=−63−6=−69
Step 3
x=7−69
Final Answer
x=−769
Question 5
Step 1
8x+81−4x=7
Multiply both sides by 8:
8×8x+1−4x=56
64x+1−4x=56
60x+1=56
Step 2
60x=56−1=55
Step 3
x=6055=1211
Final Answer
x=1211
Question 6
Step 1
x5=−6
Multiply both sides by x (assuming x=0):
5=−6x
Step 2
x=−65=−65
Final Answer
x=−65
Question 7
Step 1
4×11+9=3
Subtract 9 from both sides:
4×11=3−9=−6
Multiply both sides by 4x
11=−6×4x=−24x
Step 2
x=−2411=−2411
Final Answer
x=−2411
Question 8
Step 1
5−43x=−8x
Add 8x to both sides:
5−43x+8x=0
5+(8x−43x)=0
8x−43x=432x−43x=429x
5+429x=0
Step 2
429x=−5
Step 3
x=−5×294=−2920
Final Answer
x=−2920
Question 9
Step 1
2+34x=7−1
7−1=6, so:
2+34x=6
Step 2
34x=6−2=4
Step 3
4x=12⟹x=3
Final Answer
x=3
Question 10
Step 1
4−23x=−3x+5
Add 3x to both sides:
4−23x+3x=5
4+(3x−23x)=5
3x−23x=26x−23x=23x
4+23x=5
Step 2
23x=5−4=1
Step 3
x=32
Final Answer
x=32
Question 11
Step 1
6−x2=−10
Bring 6 to other side:
−x2=−10−6=−16
Multiply both sides by −1:
x2=16
Step 2
2=16x
Step 3
x=162=81
Final Answer
x=81
Question 12
Step 1
4−92x+x=−1
Bring all terms to one side:
4+x−92x+1=0
5+x−92x=0
x−92x=99x−92x=97x
So,
5+97x=0
Step 2
97x=−5
Step 3
x=−5×79=−745
Final Answer
x=−745
Section B
Concepts
Simplifying Algebraic Expressions, Distributive Law, Combining Like Terms, Solving Linear Equations
Question 1
4(2x−3)−8(2x+5)
Step 1
4×2x−4×3−8×2x−8×5
8x−12−16x−40
Step 2
8x−16x−12−40=−8x−52
Final Answer
−8x−52
Question 2
3(4x−5)−5(2x−5)
Step 1
3×4x−3×5−5×2x+5×5
12x−15−10x+25
Step 2
12x−10x−15+25=2x+10
Final Answer
2x+10
Question 3
8(6x+2)−5(x−2)
Step 1
8×6x+8×2−5x+10
48x+16−5x+10
Step 2
48x−5x+16+10=43x+26
Final Answer
43x+26
Question 4
2(3x−4)−7(11−2x)
Step 1
2×3x−2×4−7×11+7×2x
6x−8−77+14x
Step 2
6x+14x−8−77=20x−85
Final Answer
20x−85
Question 5
7(5−x)=−4(x−11)
Step 1
35−7x=−4x+44
Step 2
35−44=−4x+7x
−9=3x
Step 3
x=−3
Final Answer
x=−3
Question 6
−4(x−8)=−6(4+3x)
Step 1
−4x+32=−24−18x
Step 2
−4x+32+18x+24=0
14x+56=0
Step 3
14x=−56⟹x=−4
Final Answer
x=−4
Question 7
7(4−3x)−2(8x−9)+6
Step 1
28−21x−16x+18+6
Step 2
−21x−16x+28+18+6
−37x+52
Final Answer
−37x+52
Question 8
−6(3−4x)+2x−8(x+11)
Step 1
−18+24x+2x−8x−88
24x+2x−8x=18x
−18−88=−106
Total: 18x−106
Final Answer
18x−106
Question 9
3(2x−6)−3−4(3−x)
Step 1
6x−18−3−12+4x
Step 2
6x+4x−18−3−12=10x−33
Final Answer
10x−33
Question 10
9(2x−1)−3x−3(12+x)
Step 1
18x−9−3x−36−3x
18x−3x−3x=12x
−9−36=−45
Total: 12x−45
Final Answer
12x−45
Question 11
4x−(2x−8)=5(1+2x)
Step 1
4x−2x+8=5+10x
2x+8=5+10x
Step 2
2x−10x=5−8
−8x=−3
Step 3
x=−8−3=83
Final Answer
x=83
Question 12
10−6(8x−2)−9x−(3+4x)
Step 1
10−48x+12−9x−3−4x
10+12−3=19
−48x−9x−4x=−61x
Total: 19−61x
Final Answer
19−61x
Section C
Concepts
Equating Rational Expressions, Cross Multiplication, Linear Equations
Question 1
(5x−2)/3=(4x+1)/2
Step 1
Cross-multiply: 2(5x−2)=3(4x+1) 10x−4=12x+3
Step 2
10x−12x=3+4 −2x=7 x=−27
Final Answer
x=−27
Question 2
(7x−8)/5=(2x+5)/4
Step 1
4(7x−8)=5(2x+5) 28x−32=10x+25
Step 2
28x−10x=32+25 18x=57 x=1857=619
Final Answer
x=619
Question 3
2−8x−1=65−3x
Step 1
Cross-multiply: 6(−8x−1)=2(5−3x) −48x−6=10−6x
Step 2
−48x+6x=10+6 −42x=16 x=−4216=−218
Final Answer
x=−218
Question 4
35(x+11)=23(1+x)
Step 1
Cross-multiplication: 2×5(x+11)=3×3(1+x) 10(x+11)=9(1+x) 10x+110=9+9x
Step 2
10x−9x=9−110 x=−101
Final Answer
x=−101
Question 5
43(2+5x)=52(6x−3)
Step 1
Cross-multiply: 5×3(2+5x)=4×2(6x−3) 15(2+5x)=8(6x−3) 30+75x=48x−24
Step 2
75x−48x=−24−30 27x=−54 x=−2
Final Answer
x=−2
Question 6
32(3x−5)=74(x−2)
Step 1
Cross-multiply: 7×2(3x−5)=3×4(x−2) 14(3x−5)=12(x−2) 42x−70=12x−24
Step 2
42x−12x=−24+70 30x=46 x=3046=1523
Final Answer
x=1523
Question 7
21(2x−6)=41(8−12x)
Step 1
Simplify both sides: 1(x−3)=2−3x
x−3=2−3x
Step 2
x+3x=2+3 4x=5 x=45
Final Answer
x=45
Question 8
21(5x+7)=43(3x−1)
Step 1
25x+7=49x−3
Cross-multiply: 4(5x+7)=2(9x−3) 20x+28=18x−6
Step 2
20x−18x=−6−28 2x=−34 x=−17
Final Answer
x=−17
Question 9
3x+15=12
Step 1
Cross-multiply: 5=12(3x+1)
5=36x+12
Step 2
36x=5−12=−7 x=−367
Final Answer
x=−367
Question 10
x+3x+2=4
Step 1
Cross-multiply: x+2=4(x+3) x+2=4x+12
Step 2
x−4x=12−2 −3x=10 x=−310
Final Answer
x=−310
Question 11
3x−22x−9=−3
Step 1
Cross-multiplied: 2x−9=−3(3x−2)=−9x+6
Step 2
2x+9x=6+9 11x=15 x=1115
Final Answer
x=1115
Question 12
3x+102=x−11
Step 1
Cross-multiply: 2(x−1)=3x+10 2x−2=3x+10
Step 2
2x−3x=10+2 −x=12 x=−12
Final Answer
x=−12
Question 13
27x+3=92x−5
Step 1
Cross-multiply: 9(7x+3)=2(2x−5) 63x+27=4x−10
Step 2
63x−4x=−10−27 59x=−37 x=−5937
Final Answer
x=−5937
Question 14
6x+128=7x−1011
Step 1
Cross-multiplied: 8(7x−10)=11(6x+12) 56x−80=66x+132
Step 2
56x−66x=132+80=212 −10x=212 x=−10212=−5106
Final Answer
x=−5106
Shorter Method ?
Efficient Approaches for Solving Worksheet Linear Equations
Section A: Single-Variable Fractional Equations
- When equation is of the form cax+b=d:
- Immediately write: ax+b=c×d, then x=acd−b
- This skips the writing-out multiplications; the answer can be computed in a single mental step.
Example:
- 32x+5=11 Rightarrow 2x+5=33 Rightarrow x=233−5=14
- 65−9x=−2 Rightarrow 5−9x=−12 Rightarrow x=−95+12=−917
- For bx+ca=d, rearrange to a=d(bx+c), solve for x instantly.
Section B: Expand Quickly, Group Quickly
- For all k(mx+n)+p(qx+r) forms, directly expand all x terms and constants mentally, then group all x, group all constants:
- 4(2x−3)−8(2x+5):
- =4×2x−4×3−8×2x−8×5
- =8x−12−16x−40=(8x−16x)+(−12−40)=−8x−52
- Do the same for every Section B problem, skipping intermediary written steps.
Section C: Cross Multiplication Shortcuts
- For any equation of the form cax+b=rpx+q:
- Instantly write: (ax+b)×r=(px+q)×c
- Collect x terms and constants: arx−pcx=qc−br
- x=ar−pcqc−br
- By memorizing/spotting this pattern, most equations in Section C can be solved nearly mentally.
Example:
- (5x−2)/3=(4x+1)/2:
- 2(5x−2)=3(4x+1)
- 10x−4=12x+3
- 10x−12x=3+4⇒−2x=7⇒x=−27
Universal Speedups:
- Always group all x terms on one side, constants on the other in a single step.
- When results are fractions, keep the calculation as x=(difference of constants)/(difference of coefficients).
For equations like bx+ca=d and cax+b=d:
- bx+c=da or ax+b=cd. Solve for x directly.
For expressions with distributed multiplication:
- Expand one bracket at a time mentally (not writing each step),
- Immediately sum/subtract like terms.
Summary Table: Pattern-Based Fast Solving
