Practice Questions

Define the term 'equation' in the context of algebra.

Explain the fundamental difference between an algebraic expression and an algebraic equation. Provide one example of each.

Solve for $p$: $2(p + 6) = 22$

Solve for $m$: $15 - 2m = 3$

Justify why it is often a preferred strategy to multiply an equation with fractional coefficients by the LCM of the denominators before solving.

In the equation $5x - 3 = 12$, what do LHS and RHS stand for? Identify the LHS and RHS in this specific equation.

Justify why the equation $x + 5 = x - 3$ has no solution, without attempting to solve it numerically.

Solve the equation: $x = \frac{3}{4}(x + 12)$

Identify the variable terms and the constant terms in the equation $9y - 15 = 4y + 5$.

Identify the linear equation in one variable from the following options: (a) $x^2 + 5 = 14$ (b) $3y - 7 = 2y + 1$ (c) $2x + 3y = 10$ (d) $z + z^3 = 2$

Solve for $x$: $5x - 9 = 2x + 3$

Solve for $x$: $8x + 4 = 4(x + 5)$

Solve for $z$: $\frac{z}{3} + 5 = 8$

Summarize the step-by-step procedure to find the solution for an equation with variables on both sides, using $4x + 7 = 2x + 15$ as an example to explain each step.

List the key characteristics that define a 'linear equation in one variable'.

Solve the equation: $\frac{3y - 4}{2} - \frac{2y + 5}{3} = 1$

Recall what is meant by the 'solution' of an equation.

List two examples of expressions that are linear and two examples of expressions that are not linear.

A student simplifies the equation $\frac{2}{3}(x+6) = 10$ to $2(x+6) = 30$ as their first step. Evaluate if this is a valid step and justify your reasoning.

Describe what it means to 'balance an equation'. Why is it important to perform the same mathematical operation on both sides?

Solve for $a$: $\frac{2a}{5} + 3 = \frac{a}{2} + 4$

Explain the method of 'transposition' for solving equations. Use the equation $5x + 8 = 23$ to illustrate the process.

Define a 'linear expression' and a 'non-linear expression'. Provide three distinct examples of each and explain for each non-linear example why it does not qualify as linear.

Explain the full process for solving an equation that includes brackets, such as $5(y - 3) = 3(y + 1)$. Your explanation should describe each general step and how it applies to the example.

Calculate the value of $x$ for the equation: $0.5(x - 1) = 0.2(x + 8)$

The length of a rectangle is 5 cm more than its width. If the perimeter of the rectangle is 74 cm, analyze the problem to find its length and width.

The sum of two numbers is 85. If one number exceeds the other by 15, analyze the statement and find the numbers.

The present age of Sahil’s mother is three times the present age of Sahil. After 5 years, their ages will add to 66 years. Calculate their present ages.

A piggy bank contains only Rs. 2 and Rs. 5 coins. The number of Rs. 2 coins is double the number of Rs. 5 coins. If the total money in the piggy bank is Rs. 180, calculate the number of each type of coin.

Design a word problem regarding the perimeter of a rectangle where the length is defined in terms of its breadth. The formulation of the problem must lead to the linear equation $2((2b-5) + b) = 50$, where 'b' represents the breadth. Also, solve for the dimensions.

Propose a method to solve the equation $0.25(4f - 3) = 0.05(10f - 9)$ without working with decimals. Justify your method and then solve the equation.

Solve the equation: $\frac{x-2}{4} = \frac{x+1}{3}$

Critique the following statement and provide a valid reason: 'An equation like $x^2 - 4 = (x-2)(x+2)$ is a linear equation because when simplified, the variable $x$ disappears.'

Critique the following solution step: From $5x = 25$, the next step is written as $x = 25 - 5$. Identify the error and explain the correct principle.

Create a linear equation in one variable that includes parentheses on one side and a fraction on the other, such that the solution is $x = -2$.

Formulate a linear equation for the following scenario and solve it: 'The sum of three consecutive odd integers is 57. Find the integers.'

The sum of the digits of a two-digit number is 9. When the digits are interchanged, the resulting new number is 27 greater than the original number. Analyze the conditions and find the original two-digit number.

Summarize the general strategy for solving a linear equation that contains fractions, such as $\frac{x-2}{3} + 1 = \frac{x}{2}$. Explain the purpose of each major step.

Design a linear equation of the form $\frac{ax+b}{c} = \frac{dx+e}{f}$ where $a, b, c, d, e, f$ are non-zero integers, such that the solution is $x = \frac{1}{2}$. Justify that your created equation yields this solution.

A steamer goes downstream from one port to another in 9 hours. It covers the same distance upstream in 10 hours. If the speed of the stream is 1 km/hr, formulate an equation to find the speed of the steamer in still water and solve it. Also find the distance between the ports.

For the equation $\frac{z}{3} + 5 = \frac{1}{2}$, identify the first operation you would perform to simplify it by clearing the denominators and explain why this step is useful.

The sum of the digits of a two-digit number is 9. If the digits are reversed, the new number is 27 greater than the original number. Formulate an equation for this problem, justifying your choice of variable representation, and solve for the original number.

A student solved the equation $\frac{x-2}{4} - \frac{x-3}{6} = \frac{1}{3}$. Their work is shown below. Evaluate each step, identify the first error, and provide the correct solution. Student's work: Step 1: $3(x-2) - 2(x-3) = 4$ Step 2: $3x - 6 - 2x + 6 = 4$ Step 3: $x = 4$