Solving Linear Equations: math practice, Grade 9 – download the PDF

1 Evaluating expressions ★★★

Evaluate each expression for the given value.

1. \(3x+5\) for \(x=4\)
2. \(20-2x\) for \(x=6\)
3. \(x^2+x\) for \(x=5\)
4. \(4(x-1)\) for \(x=-2\)

2 Order of operations ★★★

Calculate, following the order of operations.

1. \(6+4\cdot 5\)
2. \((6+4)\cdot 5\)
3. \(18-12\div 4\)
4. \(3+2(8-5)^2\)

3 From words to algebra ★★★

Write an algebraic expression for each phrase.

1. Seven more than twice a number \(n\).
2. The product of 5 and the difference of \(m\) and 3.
3. A number \(k\) decreased by 9, then divided by 2.

4 One-step equations ★★★

Solve and check.

1. \(x+9=15\)
2. \(y-7=12\)
3. \(6z=42\)
4. \(\dfrac{n}{4}=-5\)

5 Name the property ★★★

Name the property of equality used in each step.

1. From \(x-6=10\) to \(x=16\).
2. From \(3x=21\) to \(x=7\).
3. From \(\dfrac{x}{5}=4\) to \(x=20\).
4. From \(x+8=3\) to \(x=-5\).

6 Two-step equations ★★★

Solve each equation.

1. \(3x+4=19\)
2. \(4y-6=10\)
3. \(\dfrac{x}{3}+2=7\)
4. \(10-2k=4\)

7 Gym membership ★★★

A gym charges a $15 sign-up fee plus $12 per month. Maya has paid $87 in total. Write an equation and find how many months she has been a member.

8 Solution on a number line ★★★

Solve \(2x+9=3\), then explain where the solution is located on a number line.

9 Distributing first ★★★

Solve each equation.

1. \(3(x+2)=24\)
2. \(2(3x-1)=28\)
3. \(5(x-3)+2=22\)

10 Combining like terms ★★★

Simplify each side, then solve.

1. \(4x+3-x+9=30\)
2. \(7y-2y+5=-10\)
3. \(8-3n+n=20\)

11 Fractions in equations ★★★

Solve each equation.

1. \(\dfrac{x}{2}+\dfrac{x}{3}=10\)
2. \(\dfrac{2x-1}{3}=5\)
3. \(\dfrac{3}{4}x-2=10\)

12 Decimals in equations ★★★

Solve each equation.

1. \(0.6x+1.5=4.5\)
2. \(2.4-0.5y=0.9\)

13 Variables on both sides ★★★

Solve each equation and check your answer.

1. \(5x+2=3x+12\)
2. \(4n-7=2n+9\)
3. \(6-k=3k-10\)

14 Find the error ★★★

A student solved \(3(x-4)=2x+5\) like this:

\(3x-4=2x+5\), then \(x-4=5\), so \(x=9\).

Explain the mistake and give the correct solution.

15 Perimeter of a rectangle ★★★

A rectangle has length \((2x+3)\) cm and width \(x\) cm. Its perimeter is 36 cm.

Find \(x\), then the length and the width.

16 Rearranging formulas ★★★

1. Solve \(P=2l+2w\) for \(w\), then find \(w\) when \(P=50\) and \(l=15\).
2. Solve \(F=1.8C+32\) for \(C\), then find \(C\) when \(F=86\).
3. Solve \(y=mx+b\) for \(x\) (with \(m\neq 0\)), then find \(x\) when \(y=11\), \(m=2\) and \(b=-3\).

17 A general equation ★★★

Solve \(ax+b=cx+d\) for \(x\), assuming \(a\neq c\). Then use the result with \(a=7\), \(b=1\), \(c=3\), \(d=13\).

18 How many solutions? ★★★

Decide whether each equation has one solution, no solution or infinitely many solutions. Solve it when there is exactly one.

1. \(4(x+1)=4x+7\)
2. \(3x-6=3(x-2)\)
3. \(2x+5=5x-4\)
4. \(6(x-1)-2x=4x-6\)

19 Choosing a parameter ★★★

Consider the equation \(3x+k=3(x+4)\), where \(k\) is a number.

1. For which value of \(k\) does the equation have infinitely many solutions?
2. For which values of \(k\) does it have no solution?

20 Bike rental ★★★

Shop A rents a bike for $8 plus $4 per hour. Shop B charges $20 plus $2.50 per hour. After how many hours do both shops charge the same amount, and what is that amount?

21 Consecutive odd integers ★★★

The sum of three consecutive odd integers is 87. Find the three integers.

22 Fractions on both sides ★★★

Solve \(\dfrac{2}{3}(x-6)=\dfrac{1}{2}x-1\).