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

1 Evaluating expressions ★★★

1. \(3\cdot 4+5=12+5=17\).
2. \(20-2\cdot 6=20-12=8\).
3. \(5^2+5=25+5=30\).
4. \(4(-2-1)=4\cdot(-3)=-12\).

2 Order of operations ★★★

1. Multiply first: \(6+20=26\).
2. Parentheses first: \(10\cdot 5=50\).
3. Divide first: \(18-3=15\).
4. Parentheses, then exponent, then multiplication: \(3+2\cdot 3^2=3+2\cdot 9=3+18=21\).

3 From words to algebra ★★★

1. Twice \(n\) is \(2n\), and seven more gives \(2n+7\).
2. The difference of \(m\) and 3 is \(m-3\); the product with 5 is \(5(m-3)\).
3. Decreased by 9 gives \(k-9\), then divided by 2 gives \(\dfrac{k-9}{2}\).

4 One-step equations ★★★

1. Subtract 9: \(x=6\). Check: \(6+9=15\).
2. Add 7: \(y=19\). Check: \(19-7=12\).
3. Divide by 6: \(z=7\). Check: \(6\cdot 7=42\).
4. Multiply by 4: \(n=-20\). Check: \(\dfrac{-20}{4}=-5\).

5 Name the property ★★★

1. Addition Property of Equality (add 6 to both sides).
2. Division Property of Equality (divide both sides by 3).
3. Multiplication Property of Equality (multiply both sides by 5).
4. Subtraction Property of Equality (subtract 8 from both sides: \(3-8=-5\)).

6 Two-step equations ★★★

1. \(3x=15\), so \(x=5\).
2. \(4y=16\), so \(y=4\).
3. \(\dfrac{x}{3}=5\), so \(x=15\).
4. \(-2k=-6\), so \(k=3\).

7 Gym membership ★★★

Let \(m\) be the number of months. The total cost is \(12m+15=87\). Subtract 15: \(12m=72\). Divide by 12: \(m=6\).

Check: \(12\cdot 6+15=87\). Maya has been a member for 6 months.

8 Solution on a number line ★★★

Subtract 9: \(2x=-6\). Divide by 2: \(x=-3\). Check: \(2(-3)+9=3\).

The solution is three units to the left of 0.

9 Distributing first ★★★

1. \(3x+6=24\), so \(3x=18\) and \(x=6\).
2. \(6x-2=28\), so \(6x=30\) and \(x=5\).
3. \(5x-15+2=22\), so \(5x-13=22\), \(5x=35\) and \(x=7\).

10 Combining like terms ★★★

1. \(3x+12=30\), so \(3x=18\) and \(x=6\).
2. \(5y+5=-10\), so \(5y=-15\) and \(y=-3\).
3. \(8-2n=20\), so \(-2n=12\) and \(n=-6\).

11 Fractions in equations ★★★

1. Multiply by 6: \(3x+2x=60\), so \(5x=60\) and \(x=12\).
2. Multiply by 3: \(2x-1=15\), so \(2x=16\) and \(x=8\).
3. Add 2: \(\dfrac{3}{4}x=12\). Multiply by \(\dfrac{4}{3}\): \(x=16\).

12 Decimals in equations ★★★

1. \(0.6x=3\), so \(x=\dfrac{3}{0.6}=5\).
2. \(-0.5y=-1.5\), so \(y=3\).

13 Variables on both sides ★★★

1. \(2x+2=12\), so \(2x=10\) and \(x=5\). Check: \(27=27\).
2. \(2n-7=9\), so \(2n=16\) and \(n=8\). Check: \(25=25\).
3. \(6=4k-10\), so \(4k=16\) and \(k=4\). Check: \(2=2\).

14 Find the error ★★★

The student multiplied only the first term by 3. The correct distribution is \(3(x-4)=3x-12\).

Then \(3x-12=2x+5\), so \(x-12=5\) and \(x=17\). Check: \(3(17-4)=39\) and \(2\cdot 17+5=39\).

With \(x=9\) the sides would be \(15\) and \(23\), which confirms that 9 is wrong.

15 Perimeter of a rectangle ★★★

The perimeter is \(2(2x+3+x)=36\). Simplify: \(2(3x+3)=36\), so \(6x+6=36\) and \(x=5\).

The width is 5 cm and the length is \(2\cdot 5+3=13\) cm. Check: \(2(13+5)=36\).

16 Rearranging formulas ★★★

1. \(w=\dfrac{P-2l}{2}\). With the values: \(w=\dfrac{50-30}{2}=10\).
2. \(C=\dfrac{F-32}{1.8}\). With \(F=86\): \(C=\dfrac{54}{1.8}=30\).
3. \(x=\dfrac{y-b}{m}\). With the values: \(x=\dfrac{11+3}{2}=7\).

17 A general equation ★★★

Subtract \(cx\) and \(b\): \(ax-cx=d-b\). Factor: \(x(a-c)=d-b\). Divide by \(a-c\) (not zero): \(x=\dfrac{d-b}{a-c}\).

With the values: \(x=\dfrac{13-1}{7-3}=3\). Check: \(7\cdot 3+1=22\) and \(3\cdot 3+13=22\).

18 How many solutions? ★★★

1. \(4x+4=4x+7\) gives \(4=7\), false: no solution.
2. \(3(x-2)=3x-6\), so both sides are identical: infinitely many solutions.
3. \(9=3x\), so \(x=3\): one solution. Check: \(11=11\).
4. The left side is \(6x-6-2x=4x-6\), identical to the right side: infinitely many solutions.

19 Choosing a parameter ★★★

The right side is \(3x+12\). Subtracting \(3x\) leaves \(k=12\).

1. If \(k=12\), the statement \(12=12\) is always true: infinitely many solutions.
2. If \(k\neq 12\), the statement \(k=12\) is false: no solution. For example, \(k=5\) gives \(5=12\).

20 Bike rental ★★★

Let \(h\) be the number of hours: \(8+4h=20+2.5h\). Subtract \(2.5h\): \(8+1.5h=20\). Subtract 8: \(1.5h=12\). Divide: \(h=8\).

Cost: \(8+4\cdot 8=40\) and \(20+2.5\cdot 8=40\). Both shops charge $40 for 8 hours.

21 Consecutive odd integers ★★★

Let the integers be \(n\), \(n+2\), \(n+4\). Then \(n+(n+2)+(n+4)=87\), so \(3n+6=87\) and \(3n=81\), thus \(n=27\).

The integers are 27, 29 and 31. Check: \(27+29+31=87\).

22 Fractions on both sides ★★★

Multiply every term by 6: \(4(x-6)=3x-6\). Distribute: \(4x-24=3x-6\). Subtract \(3x\): \(x-24=-6\). Add 24: \(x=18\).

Check: \(\dfrac{2}{3}\cdot 12=8\) and \(\dfrac{1}{2}\cdot 18-1=8\).