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Solved problems for the chapter:

One-step Linear Equations

View the theory »

Integers for KS3

1. Presenting integers on a number line

2. Arithmetic operations of integers

2.1. Subtraction of Integers

2.2. Properties of arithmetic operations

2.2.1. Existence of an identity element

2.2.2. The sum of any integer and its additive inverse

2.2.3. The negation of an additive inverse is its positive value

2.2.4. The additive inverse of a sum

2.2.5. Identity element for multiplication

2.2.6. Multiplication by (-1)

2.2.7. Multiplication by 0

2.2.8. The product of positive and negative integers

2.2.9. The product of a negative and a positive integer

3. Removing brackets

4. Order of integers

4.1. Greater than " > "

4.2. Less than " < "

4.3. Equals to " = "

4.4. Less than or equal to "

4.5. Greater than or equal to " >= "

4.5.1. Properties of "Less than"

4.5.2. Properties of "less than or equal to"

4.5.3. Properties of "greater than or equal to"

5. Forum

↓ Show theory [1] ↓

Detailed solution steps are available at:

https://en.openprof.com/wb/chapter:one-step_linear_equations/4346/

Example 1.

Olivia is 14 years old. She has a brother, Luke, who is 8 years younger than her. Mom told her she would get new roller skates when she was twice as old as Luke. How many more years does Olivia have to wait for new roller skates?

Example 2.

Petra's mother is 6 times older than Petra. They are sum of their ages is 35 years. How old is Petra and her mother?

Ivra met Tom at sea. She asked him how old he was. He replied, "I was a third as I am today 10 years ago." How old is Tom?

Daisy's great-grandmother is 10 times older than Daisy today, and in 5 years she will be 7 times older than Daisy. How old will Daisy be in 5 years' time? How old is her great-grandmother today?

How many years ago was Ned's father 8 times older than his son, if they are 10 and 45 years old today?

Oliver today he is 6 times older than his puppy Tara. In 8 years, the puppy will be half of Sam's age. How old will Sam and Tara the puppy be in 8 years?

Nancy is 3 times older than her sister Zara. She was 5 times older than her 2 years ago. The age of their brother John is 1 higher than the sum of Nancy and Zara's age 2 years ago. How old are Nancy, Zara and John?

Grandma and grandpa are both 65 years old. Their grandchildren are: Ana 2 years, Ben 3 years, Chris 5 years, Eva 8 years and Jan 10 years. In how many years will the sum of their ages be 4 times the age of all the grandchildren combined?

Aron and John are 24 years old together. In 3 years, John will be 2 times older than Aron. How old will Aron and John be in 3 years?

10.

The Kelly's family consists of four family members: mother, father, son Gal and daughter Ana. Dad and Mom are 74 years old together. Gal’s age is equal to one-sixth of her mother’s age, and Ana’s age is 7 years less than half her father’s age. How old are the members of the Kelly's family if the sum of their ages is equal to 92?

11.

Solves the inequality for rational numbers and mark the solution on the number line:

12.

Solve the inequality in the set . Mark the solution on the number line:

13.

Find the natural numbers x for which:

For (d.) find all other integers x that correspond to the inequality.

14.

Solve the set inequality . Write down the elements that make up the solutions and do a test.

15.

For which the inequality holds. Write down a set of solutions.

16.

Solve the inequality for . Write a set of solutions.

17.

Solve the inequality for . Also check the solution.

18.

Which natural numbers satisfy the inequality?

19.

Solve the inequality.

20.

Solve the inequality.

21.

Solve the inequality.

22.

What is the solution to the given inequality?

23.

Solve the inequality.

24.

For what values of x does the inequality apply?

25.

Write down the set of solutions to the inequality:

26.

Write down the set of solutions of the inequality.

27.

Show the set of solutions of the inequality on a number line:

28.

Which numbers solve the inequality.

29.

Draw a set of solutions on the number line that solves the equation .

30.

Solve the inequality for .

31.

Write an inequality whose solutions are represented on the number line.

32.

For what natural numbers does the inequality apply?

33.

Solve the inequality and show its solution on the number line:

34.

Solve the equation depending on the parameter .

35.

Solve the equation depending on the parameter .

36.

Solve the equation depending on the parameter .

37.

Evaluate:

38.

For the given values of the variables, calculate the value of the expression:

; for

39.

Solve the simultaneous equations using substitution method:

40.

If the length of a rectangle is 2a and the width is twice as short as its length. Calculate:

the perimeter
and the area

41.

Two sides of a square of side length 5a are each increased by 3a and the other two sides are each shortened by a.

Find its new length.
Find its new width.
Calculate its perimeter.

42.

Simplify the following expressions and determine which of these expressions is a single term and determine its coefficient.

43.

Calculate the perimeter of an isosceles trapezoid with base length 4a. The length of the parallel side to the base is equal to half the base. The length of each of the two unparallel sides is a.

44.

Solve the equation:

45.

Solve the equation:

46.

Solve the simultaneous equations using elimination method:

47.

Check if is a solution to the following equations:

48.

Check if is the solution of the following equations:

49.

The area of the rectangle measures . How wide is the rectangle if it is long?

Solve the equations:

Solve the equation:

Solve the equations:

Bruce bought 4 notebooks at , 2 pencils at and 10 pens at the mall. What was the price of a pen if he spent a total of at the mall?

54.

Jan picked a number, he multiplied it by 16 and got the difference of the numbers 112 and 35.2. Which number did Jan chose?

55.

Solve the equation and make a test:

56.

Solve the equation and make a test:

57.

Text problems that are solved with linear equations, which are written with the help of available data from the problem.

Checks if the number is a solution to the equation:

58.

Solve the equation:

59.

If we subtract from the triple of a number, we get . Determine this number.

60.

One-third of a number is less than its half. What number is this?

61.

If we add to the quintuple of a number, we get the same as multiplying this number increased by by .

62.

Solve the equation and do the test:

63.

Solve the equation and do the test:

Solve the equation:

Solve the equation:

Solve the equation:

Solve the equation:

Solve the equation:

Solve the equation:

Solve the equation:

Solve the equation:

Solve the equation:

Solve the equation:

Solve the equation and do the test:

Solve the equation:

Solve the equation:

Solve the equation:

Solve the equation.

Calculate the unknown x and the perimeter of the parallelogram in the figure below. All units are in centimetres.

80.

Express the quantity as the subject of the formula:

81.

Calculate the unknown value x and the perimeter of the parallelogram in the figure. All units are in centimetres.

82.

From the formula, express the quantity as the subject:

83.

What is the length c of the ladder that Peter needs to install gutters on the house if ? Express the result in meters.

84.

Calculate the perimeter and area of an equilateral triangle if the length of the side is .

85.

Expose the quantity from the formula:

86.

The distance AB consists of three parts as shown in the sketch.

Calculate the unknown if the distance AB is 52 cm long.

87.

The algebraic expressions and have the same value. Write the appropriate equation and find the value of the variable for which the above statement applies.

88.

Expose the quantity from the form:

89.

Calculate the interior and exterior angles of a triangle in which angle is 3.5 times greater than angle and angle is 1.5 times greater than angle .

Interior and exterior angle markings

90.

Triangles ABC and ADE are similar:

The following lines are known: |AC| is , |AD| is , |AE| is and |BC| is . Calculate the lengths |AB| and |DE|. Use the appropriate proportions that apply to similar triangles to solve the problem.

91.

The angle of a triangle is equal to the sum of angle and three halves of angle . Calculate the angles of a triangle if angle is equal to four fifths of angle where are the interior angles of the triangle.

92.

The sum of the squares of the sides in a right triangle is equal to 225 and is 204 longer than the sum of the sides. Calculate the perimeter of this triangle.

93.

Calculate the perimeter and area of a right triangle in which hypotenuse a is long, one side is and the other is .

94.

Find x so that the perimeter of the triangle is . Hence find the sides of the triangle.

95.

The parallelogram ABCD is given:

Calculate its perimeter.
Finds how many times side b is longer (or shorter) than side a.

96.

One side of a right triangle is shorter than the hypotenuse, and the other side is . Calculate the perimeter and area of this triangle.

97.

The figure is given:

Calculate:

area of a right triangle,
the number of squares above the hypotenuse c, if the square is divided into smaller squares in the same way as the two squares above the legs are divided,
sides of a right triangle and
the perimeter of the triangle.

Find out the necessary information with the help of a picture. Take that the side of the square is long.

98.

The height of an isosceles triangle is shorter than the sides and longer than the base. Calculate the lengths of all three sides, the height h and the area if the perimeter is .

99.

The perimeter of a right triangle is . Calculate its area if one side is shorter than the hypotenuse by and longer than the other side by . Use a calculator to check that this triangle is indeed right-angled.