2.4 Simple Algebra, Linear Equations & Expressions

Algebra forms a vital part of the quantitative reasoning landscape. For many candidates taking the Nigeria Police Exam, algebra can seem abstract and intimidating. However, at its core, algebra is simply a way of representing unknown quantities using letters (variables) and applying logical rules to solve for those unknowns. This comprehensive section will break down algebraic expressions, equations, and translation techniques in immense detail, ensuring you build the robust skills required for success.

Systematic Steps for Solving Linear Equations

• Step 1: Apply Distributive Law: Remove brackets by expanding $a(b + c) = ab + ac$.
• Step 2: Combine Like Terms: Group terms with identical variable components.
• Step 3: Isolate the Variable: Perform inverse operations on both sides to balance the equation.
• Step 4: Solve and Verify: Divide by the variable's coefficient and check the solution in the original equation.

Algebraic Expressions and Like Terms

An algebraic expression is a combination of numbers, variables, and operation symbols. For example, $3x + 5y - 2$ is an expression. The parts of an expression separated by addition or subtraction signs are called 'terms'. In our example, the terms are $3x$, $5y$, and $-2$. The number multiplied by a variable is known as the 'coefficient'. The coefficient of $x$ is 3, and the coefficient of $y$ is 5. A term without a variable, like $-2$, is called a 'constant'.

A fundamental skill in algebra is simplifying expressions by 'combining like terms'. Like terms are terms that contain the exact same variables raised to the exact same powers. The coefficients do not need to match. For instance, $4x$ and $-7x$ are like terms. $2x^2$ and $5x^2$ are like terms. However, $3x$ and $3y$ are not like terms, nor are $4x$ and $4x^2$. To combine like terms, you add or subtract their coefficients while keeping the variable part unchanged. Consider the expression: $7a + 4b - 3a + 2b + 5$. Group the like terms together: $(7a - 3a) + (4b + 2b) + 5$. Combine the coefficients: $4a + 6b + 5$. The expression is now fully simplified because no further like terms exist to combine.

Solving Linear Equations

An equation is a mathematical statement asserting that two expressions are equal, indicated by an equal sign ($=$). A linear equation is an equation where the highest power of the variable is 1. The goal of solving a linear equation is to isolate the variable on one side of the equal sign to determine its numerical value. The golden rule of solving equations is balance: whatever operation you perform on one side of the equation, you must perform the exact same operation on the other side.

Let us solve a multi-step linear equation systematically: $5x - 7 = 3x + 9$. First, we want to gather all terms containing the variable '$x$' on one side. We can achieve this by subtracting $3x$ from both sides: $5x - 3x - 7 = 3x - 3x + 9$, which simplifies to $2x - 7 = 9$. Next, we need to move the constant term away from the variable side. Add 7 to both sides: $2x - 7 + 7 = 9 + 7$, resulting in $2x = 16$. Finally, isolate '$x$' by dividing both sides by its coefficient, 2: $2x / 2 = 16 / 2$. This yields the solution: $x = 8$. You can always verify your answer by substituting it back into the original equation: $5(8) - 7 = 40 - 7 = 33$, and $3(8) + 9 = 24 + 9 = 33$. Since $33 = 33$, the solution is correct.

Let us examine a more complex equation involving brackets: $4(2y - 3) = 2(y + 9)$. The first step is to use the distributive property to eliminate the brackets. Distribute the 4 on the left side: $4 \times 2y - 4 \times 3 = 8y - 12$. Distribute the 2 on the right side: $2 \times y + 2 \times 9 = 2y + 18$. The equation is now $8y - 12 = 2y + 18$. Subtract $2y$ from both sides to gather variables: $6y - 12 = 18$. Add 12 to both sides to isolate the variable term: $6y = 30$. Divide by 6 to find the final value: $y = 5$.

Translating Word Problems into Algebra

One of the most challenging aspects of quantitative reasoning is translating a word problem into an algebraic equation. This requires interpreting English phrases into mathematical symbols carefully. Let us review common translation keys. 'Sum', 'more than', 'increased by' mean addition ($+$). 'Difference', 'less than', 'decreased by' mean subtraction ($-$). 'Product', 'times', 'twice' mean multiplication ($\times$). 'Quotient', 'divided by', 'ratio' mean division ($\div$ or fraction line). 'Is', 'yields', 'results in' mean equals ($=$).

Consider this problem: "The sum of three consecutive integers is 84. Find the integers." First, define the variables. Let the first integer be '$x$'. Consecutive integers follow one another sequentially, like 4, 5, 6. Therefore, the second integer is '$x + 1$', and the third is '$x + 2$'. The word "sum" tells us to add them together, and "is" means equals. The equation becomes: $x + (x + 1) + (x + 2) = 84$. Combine like terms on the left side: $3x + 3 = 84$. Subtract 3 from both sides: $3x = 81$. Divide by 3: $x = 27$. So, the first integer is 27, the second is 28, and the third is 29. Their sum is $27 + 28 + 29 = 84$, confirming our solution.

Let us analyze an age problem, a common exam trope: "A father is currently three times as old as his son. In five years, the sum of their ages will be 66. How old is the father now?" Let the son's current age be '$s$'. The father's current age is '$3s$'. In five years, the son will be '$s + 5$' and the father will be '$3s + 5$'. The problem states their sum in five years will be 66. So, $(s + 5) + (3s + 5) = 66$. Combine like terms: $4s + 10 = 66$. Subtract 10 from both sides: $4s = 56$. Divide by 4: $s = 14$. The son is currently 14 years old. The father is currently $3 \times 14 = 42$ years old.

Mastering simple algebra involves fluency in simplifying expressions, strict adherence to balance when solving equations, and meticulous translation of words into mathematical models. These skills are indispensable tools for navigating complex reasoning questions.