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Algebra - Simple Linear Equations in One Variable

Grade 7ICSE

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

🔑Concepts

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An algebraic equation is a statement of equality involving one or more variables. A 'Linear Equation in One Variable' is an equation where the highest power of the variable is 11. Visually, it represents a straight line if plotted on a coordinate plane, and on a number line, it represents a single specific point.

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The Equality Balance: Think of an equation like a physical balance scale. The '=' sign acts as the central pivot (fulcrum). For the scale to remain balanced (horizontal), any operation performed on the Left Hand Side (LHS) must also be performed on the Right Hand Side (RHS).

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Variables and Constants: In an expression like 5x+3=185x + 3 = 18, xx is the variable (the unknown value we seek), while 55, 33, and 1818 are constants. Visually, you can imagine xx as a sealed box containing a specific number of items that you need to count without opening it yet.

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The Method of Transposition: This involves moving a term from one side of the '=' sign to the other. When a term is transposed, its sign changes: +x+x becomes −x-x, −x-x becomes +x+x, ×x\times x becomes ÷x\div x, and ÷x\div x becomes ×x\times x. It is like shifting a weight from one pan of the balance to the other by changing its nature.

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Cross-Multiplication: When an equation is in the form of two fractions set equal to each other, like ab=cd\frac{a}{b} = \frac{c}{d}, we use cross-multiplication: a×d=b×ca \times d = b \times c. Visually, imagine drawing an 'X' connecting the numerator of one side to the denominator of the opposite side.

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Verification: To check if the value found for the variable is correct, substitute it back into the original equation. If the calculated value of the LHS equals the RHS, the solution is correct. This is the final step to ensure the 'balance scale' is perfectly level.

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Forming Equations: This involves converting word problems into mathematical statements. Keywords like 'is' or 'gives' represent the == sign, 'sum' represents ++, and 'difference' represents −-. Visualizing the problem as a story helps in identifying which side the operations belong to.

📐Formulae

Standard form of a linear equation: ax+b=cax + b = c

Solution for xx in standard form: x=c−bax = \frac{c - b}{a}

Rule of Addition: If x−a=bx - a = b, then x=b+ax = b + a

Rule of Subtraction: If x+a=bx + a = b, then x=b−ax = b - a

Rule of Multiplication: If xa=b\frac{x}{a} = b, then x=b×ax = b \times a

Rule of Division: If ax=bax = b, then x=bax = \frac{b}{a}

Cross-Multiplication: ax+bcx+d=mn  ⟹  n(ax+b)=m(cx+d)\frac{ax + b}{cx + d} = \frac{m}{n} \implies n(ax + b) = m(cx + d)

💡Examples

Problem 1:

Solve for xx: 4(x−3)=2(x+1)+44(x - 3) = 2(x + 1) + 4

Solution:

Step 1: Expand the brackets on both sides: 4x−12=2x+2+44x - 12 = 2x + 2 + 4 Step 2: Simplify the constants on the RHS: 4x−12=2x+64x - 12 = 2x + 6 Step 3: Transpose 2x2x to the LHS and −12-12 to the RHS: 4x−2x=6+124x - 2x = 6 + 12 Step 4: Combine like terms: 2x=182x = 18 Step 5: Divide by 22: x=182x = \frac{18}{2} x=9x = 9

Explanation:

We first use the distributive property to clear parentheses, then gather all terms containing the variable on one side and constant numbers on the other to isolate the variable.

Problem 2:

Solve the equation: 3x−24=2x+33\frac{3x - 2}{4} = \frac{2x + 3}{3}

Solution:

Step 1: Apply cross-multiplication: 3(3x−2)=4(2x+3)3(3x - 2) = 4(2x + 3) Step 2: Multiply the terms: 9x−6=8x+129x - 6 = 8x + 12 Step 3: Transpose 8x8x to the LHS and −6-6 to the RHS: 9x−8x=12+69x - 8x = 12 + 6 Step 4: Solve for xx: x=18x = 18

Explanation:

Since the equation is a proportion (two fractions equal to each other), cross-multiplication is the most efficient method to remove the denominators and turn it into a simple linear equation.