Practice
Solving Equations Practice
Fifty original questions covering the complete solving equations lesson, with explanations and SAT-focused strategy.
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Question 1
Explanation
This tests a one-step addition equation. Subtract \(7\) from both sides to preserve equality: \(x=19-7=12\). Check: \(12+7=19\).
Question 2
Explanation
Add \(8\) to both sides: \(y-8+8=-3+8\), so \(y=5\). Substitution gives \(5-8=-3\), confirming the solution.
Question 3
Explanation
The coefficient \(6\) multiplies \(n\). Divide both sides by \(6\): \(n=42/6=7\). Checking gives \(6(7)=42\).
Question 4
Explanation
Multiply both sides by \(5\): \(p=5(-4)=-20\). The check \(-20/5=-4\) preserves the negative sign.
Question 5
Explanation
Divide both sides by \(-3\): \(x=27/(-3)=-9\). A negative coefficient requires a negative solution here because \((-3)(-9)=27\).
Question 6
Explanation
Divide both sides by \(0.4\): \(t=3.2/0.4=8\). Equivalently, multiplying both sides by \(10\) gives \(4t=32\).
Question 7
Explanation
Multiply both sides by \(-7\): \(x=5(-7)=-35\). Checking \((-35)/(-7)=5\) confirms the two negative signs cancel.
Question 8
Explanation
Equality is symmetric, so the variable may appear on the right. Subtract \(4\) from both sides: \(7=m\), equivalent to \(m=7\).
Question 9
Explanation
Add \(5\) to both sides: \(-12+5=q\), so \(q=-7\). Check on the original right side: \(-7-5=-12\).
Question 10
Explanation
Multiply both sides by the reciprocal \(\frac32\): \(r=10\cdot\frac32=15\). The check \(\frac23(15)=10\) verifies the value.
Question 11
Explanation
Substitute or solve: subtract \(1\) to get \(4x=20\), then divide by \(4\) to get \(x=5\). Checking gives \(4(5)+1=21\).
Question 12
Explanation
The inverse of subtracting \(9\) is adding \(9\). Applying it to both sides gives \(x=22\). Changing only one side would destroy equality.
Question 13
Explanation
Adding \(6\) to both sides produces \(a-6+6=14+6\), so the Addition Property of Equality justifies \(a=20\).
Question 14
Explanation
Both sides are multiplied by \(8\): \(8(z/8)=8(3)\). That is the Multiplication Property of Equality, and it gives \(z=24\).
Question 15
Explanation
The variable still has coefficient \(8\). Divide both sides by \(8\): \(x=12/8=3/2\). Stopping at \(8x=12\) leaves the equation unsolved for \(x\).
Question 16
Explanation
Subtract \(5\) from both sides to obtain \(3x=24\). Divide by \(3\): \(x=8\). Check: \(3(8)+5=29\).
Question 17
Explanation
Subtract \(7\): \(-4x=24\). Divide by \(-4\): \(x=-6\). Substitution gives \(-4(-6)+7=31\).
Question 18
Explanation
Add \(5\) to both sides: \(x/3=12\). Multiply by \(3\): \(x=36\). The original equation checks because \(36/3-5=7\).
Question 19
Explanation
Subtract \(1.4\) to get \(0.6x=7.2\), then divide by \(0.6\): \(x=12\). Check: \(7.2+1.4=8.6\).
Question 20
Explanation
Add \(3\) to get \(\frac54x=15\). Multiply by \(\frac45\): \(x=15\cdot\frac45=12\). The fraction coefficient is undone by its reciprocal.
Question 21
Explanation
Divide both sides by \(2\) to get \(x+4=9\), then subtract \(4\): \(x=5\). Distribution would also work, but dividing first keeps the arithmetic shorter.
Question 22
Explanation
Divide by \(-3\): \(2x-1=-7\). Add \(1\): \(2x=-6\), then divide by \(2\): \(x=-3\). Check returns \(21\).
Question 23
Explanation
Distribute: \(4x-8+3=23\). Combine constants: \(4x-5=23\). Add \(5\) and divide by \(4\): \(x=7\).
Question 24
Explanation
Distribute the signed factor: \(5-6x-2=-15\). Combine: \(3-6x=-15\). Subtract \(3\), then divide by \(-6\): \(x=3\).
Question 25
Explanation
Combine like terms first: \(5x+9=34\). Subtract \(9\): \(5x=25\). Divide by \(5\): \(x=5\).
Question 26
Explanation
Distribute and combine: \(3x+6+2x=31\Rightarrow5x+6=31\). Subtract \(6\), then divide by \(5\): \(x=5\).
Question 27
Explanation
Use denominator \(6\): \(\frac{3x}{6}+\frac{2x}{6}=10\), so \(\frac56x=10\). Multiply by \(\frac65\): \(x=12\).
Question 28
Explanation
Multiply by \(4\): \(x-1=14\). Add \(1\): \(x=15\). Checking gives \(14/4=3.5\).
Question 29
Explanation
Multiply both sides by \(5\): \(2x+3=35\). Subtract \(3\): \(2x=32\). Divide by \(2\): \(x=16\).
Question 30
Explanation
Subtract \(1\): \((x+2)/3=5\). Multiply by \(3\): \(x+2=15\). Subtract \(2\): \(x=13\).
Question 31
Explanation
Multiply every term by the LCD \(12\): \(3x+2x=120\). Thus \(5x=120\), so \(x=24\). Substitution gives \(6+4=10\).
Question 32
Explanation
Divide by \(0.25\), or multiply by \(4\): \(x-8=20\). Add \(8\): \(x=28\).
Question 33
Explanation
Add \(0.4\): \(1.2x=6\). Divide by \(1.2\): \(x=5\). Multiplying the original by \(10\) would give the equivalent equation \(12x-4=56\).
Question 34
Explanation
Subtract \(2.5\): \(-0.5x=3.5\). Divide by \(-0.5\): \(x=-7\). Check: \(2.5-0.5(-7)=6\).
Question 35
Explanation
Multiplying both complete sides by \(3\) means every term is multiplied: \(3(x/3)+3(2)=3(7)\), so \(x+6=21\).
Question 36
Explanation
Distribution gives \((-2)(4x)+(-2)(-3)=-8x+6\). The product of two negative factors is positive, so \(-8x-6\) has a sign error.
Question 37
Explanation
Adding \(2\) only to the left changes one side's value without changing the other. Each other choice applies the same valid operation to both sides and preserves the solution set.
Question 38
Explanation
Let \(g\) be the guide count. The model is \(15+4g=47\). Subtract \(15\): \(4g=32\), then divide by \(4\): \(g=8\). The whole-number result fits the context.
Question 39
Explanation
First solve the given equation: \(3x=15\), so \(x=5\). Then evaluate the requested expression: \(6(5)+1=31\). Do not report \(x\) when a different value is requested.
Question 40
Explanation
Adding \(7\) to both sides yields \(5x=25\), which has the same solution \(x=5\). The other choices apply an operation incompletely or incorrectly.
Question 41
Explanation
Multiply every term by \(8\): \(6(x-6)+5=29\). Distribute to get \(6x-36+5=29\), so \(6x=60\) and \(x=10\).
Question 42
Explanation
Inside the bracket, \(2-(x-4)=2-x+4=6-x\). Then \(1.5(6-x)=12\), so \(9-1.5x=12\), \(-1.5x=3\), and \(x=-2\).
Question 43
Explanation
Simplify inside out: \(3(x-1)-4=3x-7\). Then \(2(3x-7)+5=27\Rightarrow6x-9=27\Rightarrow6x=36\Rightarrow x=6\).
Question 44
Explanation
Multiply by \(6\): \(3(x+1)-2(x-2)=30\). Distribute: \(3x+3-2x+4=30\), so \(x+7=30\) and \(x=23\).
Question 45
Explanation
Distribute accurately: \(0.08(50x)=4x\) and \(0.08(-25)=-2\). Thus \(4x-2=14\), so \(4x=16\) and \(x=4\).
Question 46
Explanation
Distribute \(\frac23\): \(\frac23(3x-9)=2x-6\). Because that quantity is subtracted, \(5-(2x-6)=11-2x\). Then \(11-2x=-7\), so \(x=9\).
Question 47
Explanation
The factor \(4\) must multiply every term on both sides: \(4(x/4)+4(3)=4(9)\), which gives \(x+12=36\). The student failed to multiply the constant \(3\).
Question 48
Explanation
If subtracting \(6\) is the first inverse step, the original constant was \(+6\). Dividing by \(5\) next means the variable term was \(5x\). Since \(x=4\), the original is \(5x+6=26\).
Question 49
Explanation
Length is \(2w+4\). The perimeter model is \(2w+2(2w+4)=62\). Simplifying gives \(6w+8=62\), so \(6w=54\) and \(w=9\).
Question 50
Explanation
Multiply by \(0.6\): \(0.3x-1.2=4.2\). Add \(1.2\): \(0.3x=5.4\). Divide by \(0.3\): \(x=18\). Checking the numerator gives \(4.2\), and \(4.2/0.6=7\).
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Questions to review
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- Question 1One-step equationsEasy
- Question 2One-step equationsEasy
- Question 3One-step equationsEasy
- Question 4One-step equationsEasy
- Question 5Signed coefficientsEasy
- Question 6Decimal equationsEasy
- Question 7Signed divisionEasy
- Question 8Equality symmetryEasy
- Question 9Signed additionEasy
- Question 10Fractional coefficientsEasy
- Question 11Solution checkingEasy
- Question 12Inverse operationsEasy
- Question 13Properties of equalityEasy
- Question 14Properties of equalityEasy
- Question 15Complete isolationEasy
- Question 16Two-step equationsMedium
- Question 17Two-step equationsMedium
- Question 18Two-step equationsMedium
- Question 19Decimal equationsMedium
- Question 20Fractional coefficientsMedium
- Question 21Grouped equationsMedium
- Question 22Grouped equationsMedium
- Question 23DistributionMedium
- Question 24Negative distributionMedium
- Question 25Combining like termsMedium
- Question 26Multi-step equationsMedium
- Question 27Fraction equationsMedium
- Question 28Fraction-decimal equationsMedium
- Question 29Fraction equationsMedium
- Question 30Nested inverse operationsMedium
- Question 31Clearing denominatorsMedium
- Question 32Decimal groupingMedium
- Question 33Decimal equationsMedium
- Question 34Signed decimal equationsMedium
- Question 35Clearing denominatorsMedium
- Question 36Error analysisMedium
- Question 37Equality preservationMedium
- Question 38Context equationsMedium
- Question 39Solve then evaluateMedium
- Question 40Equivalent equationsMedium
- Question 41Complex fraction equationsHard
- Question 42Nested groupingHard
- Question 43Nested groupingHard
- Question 44Complex fraction equationsHard
- Question 45Decimal distributionHard
- Question 46Fractional distributionHard
- Question 47Error analysisHard
- Question 48Reverse reasoningHard
- Question 49Geometry equationsHard
- Question 50Decimal fraction equationsHard