Practice
Linear Equations Practice
A complete SAT-style set covering equation solving, structure, solution types, modeling, tables, parameters, fractions, and numeric entry.
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Question 1
Why this choice?
This value gives \(4(4)+7=23\), not \(31\).
Why this choice?
This value gives \(4(5)+7=27\), not \(31\).
Why this choice?
Correct. Substitution gives \(4(6)+7=31\).
Why this choice?
This can result from dividing \(31\) by \(4\) before removing the constant.
Explanation
Subtract \(7\) from both sides, then divide by \(4\). The result is \(x=6\).
\(4x+7-7=31-7\)
\(4x=24\)
\(x=6\)
Question 2
Why this choice?
This can result from dividing too early and not distributing to both terms.
Why this choice?
This value does not make the original equation true.
Why this choice?
This can result from losing the \(+4\) term.
Why this choice?
Correct. Both sides equal \(25\) when \(x=6\).
Explanation
Distribute first and combine constants. The equation becomes \(6x-11=25\), so \(x=6\).
- Distribute
\(6x-15+4=25\)
- Combine like terms
\(6x-11=25\)
- Isolate the variable
\(6x=36\), so \(x=6\)
Question 3
Why this choice?
No solution would require the variables to cancel and leave a false statement.
Why this choice?
A single solution would remain only if simplifying isolated one value of \(x\).
Why this choice?
A linear equation cannot have exactly two distinct solutions.
Why this choice?
Correct. Both sides are equivalent expressions.
Explanation
Distributing produces \(8x-12=8x-12\), an identity that is true for every value of \(x\).
Question 4
Why this choice?
This does not account for the complete difference between the total and fixed fee.
Why this choice?
Correct. Ten miles cost $24, and the $5 fee brings the total to $29.
Why this choice?
This results from dividing the total by the per-mile rate and ignoring the fixed fee.
Why this choice?
This value makes the total greater than $29.
Explanation
Let \(m\) be the number of miles. Solve \(5+2.40m=29\) to get \(m=10\).
- Remove the fixed fee
\(2.40m=24\)
- Divide by the rate
\(m=24\div2.40=10\)
Question 5
Why this choice?
This is the value of \(y\) when \(x=5\), not when \(x=7\).
Why this choice?
This gives only \(3\cdot7\) and omits the intercept.
Why this choice?
Correct. The line follows \(y=3x+3\).
Why this choice?
This adds the intercept twice.
Explanation
The slope is \((15-9)/(4-2)=3\). Using either point gives \(y=3x+3\), so \(y=24\) when \(x=7\).
- Find the slope
\(m=6/2=3\)
- Find the intercept
\(9=3(2)+b\), so \(b=3\)
- Evaluate
\(y=3(7)+3=24\)
Question 6
Explanation
Subtract \(2x\) from both sides and add \(9\) to both sides. This gives \(3x=27\), so \(x=9\).
\(3x-9=18\)
\(3x=27\)
\(x=9\)
Question 7
Why this choice?
This adds \(4\) to the given value instead of doubling the entire expression.
Why this choice?
This doubles \(7x\) but does not account for the constant term correctly.
Why this choice?
This can result from doubling \(39\) and then subtracting \(4\) unnecessarily.
Why this choice?
Correct. The whole expression \(7x+4\) is doubled.
Explanation
The requested expression is exactly twice \(7x+4\). Therefore, \(14x+8=2(39)=78\).
- Match the structure
\(14x+8=2(7x+4)\)
- Use the given value
\(2(7x+4)=2(39)=78\)
Question 8
Explanation
Multiply every term by the least common denominator, \(12\). This gives \(4x+3x=168\), so \(x=24\).
- Clear the denominators
\(4x+3x=168\)
- Combine like terms
\(7x=168\)
- Divide by 7
\(x=24\)
Question 9
Why this choice?
This does not match the coefficient of \(x\) after distribution.
Why this choice?
Correct. Both sides become \(3x+6\).
Why this choice?
This matches the constant term rather than the coefficient of \(x\).
Why this choice?
This multiplies the two visible coefficients instead of matching equivalent expressions.
Explanation
The right side expands to \(3x+6\). For the equation to be true for every \(x\), the coefficient of \(x\) on the left must also be \(3\). Thus, \(k=3\).
- Expand the right side
\(3(x+2)=3x+6\)
- Match coefficients
\(kx+6=3x+6\Rightarrow k=3\)
Question 10
Why this choice?
Correct. The variable terms cancel and leave a contradiction.
Why this choice?
No value of \(x\) can make \(-5\) equal \(7\).
Why this choice?
A linear equation cannot have exactly two distinct solutions.
Why this choice?
Infinitely many solutions would leave a true identity, not a false statement.
Explanation
Distributing gives \(10x-5=10x+7\). Subtracting \(10x\) from both sides leaves the false statement \(-5=7\), so there is no solution.
Question 11
Why this choice?
Eight adult tickets would produce a total revenue of $192.
Why this choice?
Ten adult tickets would produce a total revenue of only $216.
Why this choice?
Correct. Twelve tickets of each type produce $240.
Why this choice?
Eighteen adult tickets would produce a total revenue of $312.
Explanation
Substitute \(s=12\): \(8(12)+12a=240\). Then \(12a=144\), so \(a=12\) adult tickets.
- Find student-ticket revenue
\(8(12)=96\)
- Find adult-ticket revenue
\(240-96=144\)
- Divide by the adult price
\(144\div12=12\)
Question 12
Explanation
Multiply every term by \(6\) to obtain \(2(2x-5)-3(x+1)=24\). Simplifying gives \(x-13=24\), so \(x=37\).
- Clear denominators
\(2(2x-5)-3(x+1)=24\)
- Distribute
\(4x-10-3x-3=24\)
- Combine and solve
\(x-13=24\Rightarrow x=37\)
Keyboard: use Tab to move, arrow keys to change answer choices, and Enter to check an answer.
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Questions to review
No mistakes this time. Excellent work.
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- Question 7Use equation structure efficientlyMedium
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- Question 11Interpret a linear equation in two variablesMedium
- Question 12Solve linear equations with fractionsHard