2. Linear Equations in One Variable

In the earlier classes, you have come across several algebraic expressions and equations. Some examples of expressions we have so far worked with are: 5x, 2x – 3, 3x + y, 2xy + 5, xyz + x + y + z, x2 + 1, y + y2.

Some examples of equations are: 5x = 25, 2x – 3 = 9, 6z + 10 = −2. You would remember that equations use the equality (=) sign; it is missing in expressions. Of these given expressions, many have more than one variable. For example, 2xy + 5 has two variables. We, however, restrict to expressions with only one variable when we form equations. Moreover, the expressions we use to form equations are linear. This means that the highest power of the variable appearing in the expression is 1.

This section introduces the concept of linear equations, which are equations that represent a straight line when graphed on a coordinate system. It distinguishes between algebraic expressions (like 5x or 2x - 3) and equations (such as 5x = 25) because equations have an equality sign. Linear expressions are specifically those where the highest power of the variable (e.g., x or y) is 1.

Think of linear equations like budget planning. Just as you cannot spend more than you have (you must keep the sides balanced), in linear equations, the left side must equal the right side, representing a balanced financial situation. If you earn x dollars and spend it on y items, the equation representing this situation would be linear because the highest power of your earnings (x) and items (y) is one.

These are linear expressions: 2x, 2x + 1, 3y – 7, 12 – 5z, (x – 4) + 10.

These are not linear expressions: x2 + 1, y + y2, 1 + z + z2 + z3 (since highest power of variable > 1). Here we will deal with equations with linear expressions in one variable only. Such equations are known as linear equations in one variable.

This section clarifies the difference between linear expressions and non-linear expressions. It gives examples of each, highlighting that linear expressions can only have variables to the first power (for example, 2x), while non-linear expressions include variables raised to higher powers (such as x²). The focus will remain on working with linear equations that involve only one variable.

Imagine planting seeds: if you have 5 seeds (2x can represent this) to plant, that’s straightforward. But if you have 5 seed packets each containing different quantities (like x² where x can be 1, 2, or 3) that complicates the garden plan. Linear equations help keep our gardening straightforward by using simple counts.

(a) An algebraic equation is an equality involving variables. It has an equality sign. The expression on the left of the equality sign is the Left Hand Side (LHS). The expression 2x – 3 = 7 on the right of the equality sign is the Right Hand Side (RHS). For x = 5, LHS = 2 × 5 – 3 = 7 = RHS. On the other hand, x = 10 is not a solution of the equation.

Here, we learn how to identify the 'left-hand side' (LHS) and 'right-hand side' (RHS) of an equation. Solving an equation involves finding values for the variable that make both sides equal, termed the solutions. For example, if we substitute x with 5 in the equation 2x - 3 = 7, we find that both sides equal 7, meaning x = 5 is a solution. However, if we try x = 10, we get different values for LHS and RHS, so it's not a solution.

Think of an equation as a seesaw – both sides must balance for it to work. If you put 5 kg on one side of the seesaw (LHS) and 7 kg on the other (RHS), it won't balance. If you figure out that 5 kg of items on one side matches with the 7 kg somehow, that’s your solution. Finding the right balance is key!

An equation is the equality of the values of two expressions. In the equation 2x – 3 = x + 2, the two expressions are 2x – 3 and x + 2. We now discuss how to solve such equations which have expressions with the variable on both sides.

This part talks about solving equations where variables appear on both sides. In the example 2x - 3 = x + 2, we learn to treat both sides equally and perform operations such as adding or subtracting the same number or variable term to find a solution.

Consider a balancing scale at a market. If the left side shows 2 apples minus 3 coins and the right shows one apple plus 2 coins, to know how many apples are there in total, you would shift items across the scale to keep it balanced, just like reorganizing the equation to isolate the variable.

Example 1: Solve 2x – 3 = x + 2 Solution: We have 2x = x + 2 + 3 or 2x = x + 5 or 2x – x = 5 (subtracting x from both sides). Here we subtract x from both sides to get x = 5.

This chunk walks through a practical example of solving an equation step-by-step. The solution shows how to manipulate the equation by subtracting x from both sides, simplifying it until we find that x equals 5. This method ensures we maintain the balance of the equation while isolating the variable.

Think of it as organizing a box of toys. If you have 2 cars but need to know how many more you have than 1 stuffed animal you’re putting in another box, subtracting the animals (variables) from your toy collection lets you clearly see your total cars left.

Example 16: Solve + 1 = 6x + 1 Solution: Multiplying both sides of the equation by 6. LHS becomes (2(6x + 1)) + 6 = x – 3.

In this example, we learn how to simplify complex equations using multiplication. By ensuring both sides of the equation are treated with equal operations, we can reduce the equation to a simpler form for easier solving. This process may involve combining like terms and balancing both sides.

Imagine having multiple packs of candies divided among friends. If you multiply the number of packs by how many candies are in each to see totals in hand, simplifying the equation works similarly. You'll know how many more packs are needed based on the balance of what remains.

1. An algebraic equation is an equality involving variables. 2. Linear equations in one variable have the highest power of a variable as 1. 3. Solutions of equations can be found by balancing both sides. 4. Sometimes equations need simplification before solving.

The final chunk summarizes everything discussed in the section, reinforcing key points about linear equations, the nature of solutions, and importance of simplifying equations before solving them.

Recap with a sports analogy: In a game, understanding the rules (equations), how to score points (solutions), and when to reset your strategies (simplifying) makes you a better player overall. Each part clearly fits together for you to win the game.

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