SOMATH Journal · Regents Algebra I · Part II
August 2026 Algebra 1 Regents: Part II
This August 2026 Algebra I Regents Part II walkthrough covers two-way tables, literal equations, inequalities, arithmetic sequences, and radicals. Read the original questions and answer choices, then select the blue Answer control for a step-by-step SOMATH explanation: identify the idea, show the work, and check the result.
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Questions, data, and original figures are reproduced from the August 18, 2026 Algebra I examination in the NYSED Algebra I archive, with mathematical notation and layout normalized for the web. The explanations are original SOMATH teaching solutions, not official NYSED explanations or scoring rubrics. Results were checked against the NYSED scoring key and rating guide.
Show your work and answer every requested component. A correct numerical answer with no work receives only 1 credit. Use the required method when one is specified, and follow the question-specific rating criteria.
Download the original exam (PDF)
The unchanged exam booklet supplied for this lesson, including its reference sheet. No new workbook has been generated.
Completing a two-way frequency tableQuestion 25
A survey of 200 students was taken.
It was found that 140 students play sports and 50 of the students that play sports buy their lunch.
One-third of students who do not play sports bring their lunch.
Complete the two-way frequency table below.
| Buy Lunch | Bring Lunch | Total | |
|---|---|---|---|
| Play Sports | |||
| Don’t Play Sports | |||
| Total |
Answer
Answer: Play sports: 50, 90, 140; don’t play sports: 40, 20, 60; totals: 90, 110, 200.
- There are \(200-140=60\) students who do not play sports.
- Of those 60 students, one third bring lunch: \(60/3=20\). The other \(60-20=40\) buy lunch.
- Among the 140 who play sports, 50 buy lunch, so \(140-50=90\) bring it.
- Add down each column and across each row. Both sets of totals must lead to 200.
Completed frequency table Buy Lunch Bring Lunch Total Play Sports 50 90 140 Don’t Play Sports 40 20 60 Total 90 110 200
SOMATH takeaway: Apply a fraction to the group it describes, not to the entire survey.
Rearranging a literal equationQuestion 26
Solve the formula \(y=mx+b\) for \(x\) in terms of \(y\), \(m\), and \(b\).
Answer
Answer: \(x=\frac{y-b}{m}\), for \(m\ne0\).
- Undo the addition of \(b\) by subtracting it from both sides: \(y-b=mx\).
- Undo multiplication by \(m\) by dividing both sides by \(m\): \(x=\frac{y-b}{m}\).
- The division assumes \(m\ne0\). If \(m=0\), the original equation becomes \(y=b\) and cannot determine a unique \(x\).
SOMATH takeaway: Isolate the requested variable using inverse operations, treating the other letters as constants.
Solving an inequality with fractionsQuestion 27
Solve \(\frac34x+\frac25<\frac14x-\frac25\) for \(x\).
Answer
Answer: \(x<-\frac85\).
- Multiply every term by the positive least common denominator 20: \(15x+8<5x-8\). The inequality direction stays the same.
- Subtract \(5x\), then subtract 8: \(10x<-16\).
- Divide by positive 10: \(x<-16/10=-8/5\).
- Check a value in the solution, such as \(x=-2\): the original sides are \(-1.1\) and \(-0.9\), and \(-1.1<-0.9\).
SOMATH takeaway: Reverse an inequality only when multiplying or dividing by a negative number.
Writing an arithmetic sequenceQuestion 28
Given the sequence: \(-5,-2,1,4,\ldots\)
Write an equation for the nth term of this sequence.
State the tenth term of this sequence.
Answer
Answer: \(a_n=-5+3(n-1)=3n-8\); \(a_{10}=22\).
- The sequence increases by 3 each time, so it is arithmetic with \(a_1=-5\) and common difference \(d=3\).
- Use \(a_n=a_1+(n-1)d\): \(a_n=-5+3(n-1)\).
- For the tenth term, \(n=10\): \(a_{10}=-5+3(9)=-5+27=22\).
- There are nine increases between the first and tenth terms, not ten.
SOMATH takeaway: Count the jumps from the first term: the nth term is n−1 jumps away.
Using the quadratic formulaQuestion 29
Use the quadratic formula to solve \(2x^2+3x-1=0\) for the exact values of \(x\).
Answer
Answer: \(x=\frac{-3+\sqrt{17}}4\) or \(x=\frac{-3-\sqrt{17}}4\).
- Identify \(a=2\), \(b=3\), and \(c=-1\). Keep the negative sign in \(c\).
- Use the required method: \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\).
- Substitute: \(x=\frac{-3\pm\sqrt{3^2-4(2)(-1)}}{2(2)}=\frac{-3\pm\sqrt{17}}4\).
- The radicand 17 has no square factor greater than 1. Leave both answers in radical form because exact values are requested.
SOMATH takeaway: A decimal approximation does not replace an exact answer, and both signs supply solutions.
Simplifying and combining radicalsQuestion 30
Perform the indicated operation and express your answer in simplest radical form. \[5\sqrt{32}+10\sqrt2\]
Answer
Answer: \(30\sqrt2\).
- Extract the perfect-square factor: \(\sqrt{32}=\sqrt{16\cdot2}=4\sqrt2\).
- Multiply by the coefficient: \(5\sqrt{32}=5(4\sqrt2)=20\sqrt2\).
- Now combine like radicals: \(20\sqrt2+10\sqrt2=(20+10)\sqrt2=30\sqrt2\).
SOMATH takeaway: Simplify each radical first. Then add the coefficients of matching radical parts.
Answer key
Keep the key closed until you have tried the questions. Open the individual answers above for the full reasoning.
Class review: what to remember
A short response still needs a visible chain of reasoning. State the relationship, apply the operation, and name the result. For a sequence, count the jumps; for radicals, extract perfect-square factors; for a literal equation, isolate the requested letter. Read the directions: an exact answer and a specified method are part of the task.
Quick questions and answers
Which questions are in Part II?
Part II contains Questions 25–30, worth 2 credits each, for a total of 12 credits. (NYSED scoring materials)
What is the answer to Question 28?
The nth term is a_n=−5+3(n−1), or 3n−8. The tenth term is 22.
What is the answer to Question 30?
The simplified result is 30√2. First rewrite √32 as 4√2, then combine 20√2 and 10√2.
Does Question 29 require exact answers?
Yes. Use the quadratic formula and leave the two solutions as (−3+√17)/4 and (−3−√17)/4.
Do I need to show work?
Yes. The exam requests supporting work, and a correct numerical answer without work receives only 1 credit. For other response types, follow the specific requirements in the rating guide.