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Algebra - Solving One-step and Two-step Linear Equations

Grade 7IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A variable is a letter like xx or yy that represents an unknown number, while a constant is a fixed number. In a linear equation, the variable is only raised to the power of 11. Visually, imagine xx as a closed box containing a specific number of items that we need to discover.

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The Equals Sign (==) acts as the balance point of a scale. For an equation to remain true, the 'weight' on the left side (LHS) must always equal the 'weight' on the right side (RHS). If you add 55 to one side, you must add 55 to the other to keep the scale level.

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Inverse operations are 'opposite' actions that undo each other. Addition is the inverse of subtraction, and multiplication is the inverse of division. Visually, if an equation adds 1010, we 'undo' it by moving in the opposite direction on a number line and subtracting 1010.

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Solving One-Step Equations involves performing a single inverse operation to isolate the variable. For example, in x+3=7x + 3 = 7, we subtract 33 from both sides. Visually, this is like removing 33 blocks from both sides of a balanced scale so that only the variable box remains on one side.

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Solving Two-Step Equations requires two different inverse operations, usually performed in the reverse order of operations (Reverse PEMDAS/BODMAS). We typically 'undo' addition or subtraction first, then 'undo' multiplication or division. Imagine unwrapping a gift: you remove the outer layer (the constant) before the inner layer (the coefficient).

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Verification or Substitution is the process of checking your answer. Once you find a value for the variable, plug it back into the original equation. If the LHS equals the RHS, your solution is correct. Visually, this confirms that the scale is perfectly balanced with your chosen value.

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The Coefficient is the number multiplying the variable, such as the 22 in 2x2x. To isolate xx, we divide by the coefficient. Visually, if 2x=102x = 10, we are splitting two identical boxes into two groups to see that one box must contain 55 items.

📐Formulae

One-step (Addition): x+a=b  ⟹  x=b−ax + a = b \implies x = b - a

One-step (Subtraction): x−a=b  ⟹  x=b+ax - a = b \implies x = b + a

One-step (Multiplication): ax=b  ⟹  x=baax = b \implies x = \frac{b}{a}

One-step (Division): xa=b  ⟹  x=b⋅a\frac{x}{a} = b \implies x = b \cdot a

Two-step (General Form): ax+b=c  ⟹  ax=c−b  ⟹  x=c−baax + b = c \implies ax = c - b \implies x = \frac{c - b}{a}

💡Examples

Problem 1:

Solve for xx: x−12=25x - 12 = 25

Solution:

x−12=25x - 12 = 25 x−12+12=25+12x - 12 + 12 = 25 + 12 x=37x = 37

Explanation:

To isolate xx, we identify the operation being performed on it, which is subtraction of 1212. We perform the inverse operation, which is adding 1212 to both sides of the equation to maintain balance.

Problem 2:

Solve for yy: 3y+7=223y + 7 = 22

Solution:

Step 1: Subtract 77 from both sides: 3y+7−7=22−73y + 7 - 7 = 22 - 7 3y=153y = 15

Step 2: Divide both sides by 33: 3y3=153\frac{3y}{3} = \frac{15}{3} y=5y = 5

Explanation:

This is a two-step equation. First, we remove the constant +7+7 by subtracting 77 from both sides. Then, we remove the coefficient 33 (which is multiplying yy) by dividing both sides by 33.