These 22 problems practice Common Core standards A-REI.A.1, A-REI.B.3, A-CED.A.1 and A-CED.A.4. Solve multi-step linear equations with fractions, decimals and variables on both sides, justify steps, recognize equations with no solution or infinitely many solutions, and rearrange formulas. The last six are word problems that start with writing an equation. Work from easiest to hardest.
Problem 1: Two steps with a tape diagram
Solve \(4x+9=37\).
Problem 2: Variables on both sides
Solve \(7x-3=5x+11\).
Problem 3: Distribute first
Solve \(3(x+4)=2x+19\).
Problem 4: Fractional coefficient
Solve \(\frac{2}{3}x-5=7\).
Problem 5: Fractions with a common denominator
Solve \(\frac{x}{2}+\frac{x}{3}=10\).
Problem 6: Fractions on both sides
Solve \(\frac{x+3}{4}=\frac{2x-1}{3}\).
Problem 7: Decimals on both sides
Solve \(0.4x+1.2=0.7x-0.3\).
Problem 8: Give the reasons
Maya solved \(2(x-3)+5=x+7\) in the five steps shown. Write the reason for each step after the first.
- \(2(x-3)+5=x+7\)
- \(2x-6+5=x+7\)
- \(2x-1=x+7\)
- \(x-1=7\)
- \(x=8\)
Problem 9: Find the error
Jamal solved \(3(x-2)=12\) like this: \(3x-2=12\), \(3x=14\), \(x=\frac{14}{3}\). Find his mistake and solve correctly.
Problem 10: No solution or not?
Solve \(2(x+3)=2x+5\). Explain the result.
Problem 11: Infinitely many solutions?
Solve \(4(x-2)+3=4x-5\). Explain the result.
Problem 12: Solve by graphing
The graph shows \(y=2x-1\) and \(y=-x+5\). Use it to estimate the solution of \(2x-1=-x+5\), then solve algebraically.
Problem 13: Solve for the height
The area of a triangle is \(A=\frac{1}{2}bh\). Solve for \(h\), then find \(h\) when \(A=36\) and \(b=9\).
Problem 14: Solve for width
The perimeter of a rectangle is \(P=2l+2w\). Solve for \(w\), then find \(w\) when \(P=50\) and \(l=15\).
Problem 15: Standard form to y
Solve \(2x+3y=12\) for \(y\). Then find \(y\) when \(x=3\).
Problem 16: Temperature formula
Solve \(C=\frac{5}{9}(F-32)\) for \(F\). Then convert \(35^{\circ}\text{C}\) to degrees Fahrenheit.
Problem 17: Taxi fare
Priya’s taxi ride costs \(\$3.50\) plus \(\$2.25\) per mile. The total was \(\$17.00\). Write and solve an equation for the number of miles.
Problem 18: Gym plans
Gym A charges a \(\$60\) sign-up fee and \(\$25\) per month. Gym B charges no fee and \(\$40\) per month. After how many months do both cost the same, and what is that cost?
Problem 19: Garden fence
The length of a rectangular garden is 3 feet more than twice its width. The perimeter is 54 feet. Find the dimensions.
Problem 20: Consecutive odd integers
The sum of three consecutive odd integers is 87. Find the integers.
Problem 21: Trapezoid base
The area of a trapezoid is \(A=\frac{1}{2}(b_1+b_2)h\). Solve for \(b_1\). Then find \(b_1\) when \(A=84\), \(b_2=9\) and \(h=8\).
Problem 22: Shopping with fractions
Ethan spent one third of his savings on a game and one fourth on a jacket. He has \(\$35\) left. How much did he save?
See every chapter on the Algebra 1 math page.