Formula Substitution: Replacing variables with values in expressions and formulae.

Formula Substitution. Grade 7 Math: Formula Substitution Subtopic Navigator Introduction: What is Substitution? Variables and Constants Substitution in Simple Expressions Order of Operations in Substitution Substitution with Negative Numbers Substitution in Formulas Multiple Variable Substitution Real-World Formula Applications Checking Your Substitution Practice Exercises Conclusion Learning Objectives Understand the concept of variables and constants Replace variables with given values in expressions Apply correct order of operations when substituting Substitute negative numbers correctly Use substitution in real-world formulas Substitute multiple variables in one expression Check substitution work for accuracy Apply formulas to solve practical problems Introduction: What is Substitution? Formula substitution is like following a recipe. Imagine a recipe says "Add 2 eggs" - but what if you want to double the recipe? You'd substitute "eggs" with "4 eggs". In math, we substitute variables (like x, y, or a) with actual numbers to find specific values. Key Terms: Variable: A letter or symbol that represents an unknown or changing value (e.g., x, y, n) Constant: A fixed value that doesn't change (e.g., 5, -3, 2.5) Expression: A mathematical phrase with numbers and/or variables (e.g., 3x + 2) Formula: A mathematical rule expressed using variables (e.g., A = l × w) Substitution: Replacing variables with given values Evaluate: Calculate the value after substitution Substitution Process: Expression → Replace variables → Simplify → Final answer Substitution is used everywhere in mathematics and real life: calculating area, converting temperatures, determining speed, and much more. It's a fundamental skill that bridges abstract algebra with practical calculation. Variables and Constants Before we can substitute, we need to understand what variables and constants are. Variables are like empty boxes that can hold different values, while constants are fixed numbers that never change. Identifying Variables and Constants: In the expression $3x + 5$: • $x$ is a variable (can be any number) • $3$ and $5$ are constants (always 3 and 5) • $3x$ means "3 times x" (coefficient 3, variable x) Example 1: Recognizing Variables Identify the variables and constants in each expression: 1. $4y - 7$ 2. $2a + 3b + 8$ 3. $frac{m}{2} + 5$ Solution: 1. $4y - 7$: • Variable: $y$ • Constants: $4$ (coefficient), $-7$ (constant term) 2. $2a + 3b + 8$: • Variables: $a$ and $b$ • Constants: $2$, $3$, and $8$ 3. $frac{m}{2} + 5$: • Variable: $m$ • Constants: $2$ (in denominator), $5$ (constant term) Example 2: What Can Variables Represent? Variables can represent different things in different situations: Real-world examples: • $t$ might represent time in seconds • $d$ might represent distance in meters • $P$ might represent price in naira • $n$ might represent number of people In expressions: If $x = 3$, then $2x = 2 times 3 = 6$ If $x = 5$, then $2x = 2 times 5 = 10$ The expression $2x$ changes value depending on what $x$ is. Common Mistake: Incorrect: Thinking $3x$ means "thirty-something" Correct: $3x$ means "3 times x" or "three multiplied by x" Remember: A number next to a variable means multiplication! Practice Questions In the expression $5n - 2$, what is the variable? How many variables are in the expression $2x + 3y - z$? What does $4a$ mean in words? Identify all constants in: $frac{3x}{4} + 7$ If $k$ represents kilometers, what might $5k$ represent? Substitution in Simple Expressions Substitution means replacing a variable with a given number. We do this carefully, making sure to follow all the mathematical rules. Steps for Substitution: Write the original expression Replace each variable with the given value in parentheses Simplify using order of operations (PEMDAS/BODMAS) Write the final answer Example 1: Basic Substitution Evaluate $2x + 3$ when $x = 4$ Solution: Write expression: $2x + 3$ Substitute $x = 4$: $2(4) + 3$ Multiply first: $8 + 3$ Add: $11$ Final answer: $11$ Check: When $x=4$, $2x+3 = 2times4+3 = 8+3=11$ Example 2: Substitution with Division Evaluate $frac{y}{3} + 2$ when $y = 9$ Solution: Write expression: $frac{y}{3} + 2$ Substitute $y = 9$: $frac{9}{3} + 2$ Divide first: $3 + 2$ Add: $5$ Final answer: $5$ Note: $frac{y}{3}$ means "$y$ divided by 3" or "y over 3" Example 3: Substitution with Exponents Evaluate $n^2 - 4$ when $n = 5$ Solution: Write expression: $n^2 - 4$ Substitute $n = 5$: $5^2 - 4$ Exponent first: $25 - 4$ Subtract: $21$ Final answer: $21$ Remember: $n^2$ means "$n$ squared" or "$n times n$" Expression Value of Variable Substitution Step Final Answer $3x + 2$ $x = 5$ $3(5) + 2 = 15 + 2$ $17$ $2y - 7$ $y = 4$ $2(4) - 7 = 8 - 7$ $1$ $frac{a}{2} + 3$ $a = 10$ $frac{10}{2} + 3 = 5 + 3$ $8$ $b^2 + 1$ $b = 3$ $3^2 + 1 = 9 + 1$ $10$ $4z - z$ $z = 6$ $4(6) - 6 = 24 - 6$ $18$ Principles Practice Evaluate $3x + 5$ when $x = 2$ Evaluate $2y - 4$ when $y = 7$ Evaluate $frac{m}{4} + 2$ when $m = 12$ Evaluate $k^2 + 3$ when $k = 4$ Evaluate $5n - 2n$ when $n = 3$ Order of Operations in Substitution When substituting, we must follow the order of operations (PEMDAS/BODMAS). This ensures we get the correct answer every time. Order of Operations (PEMDAS): P - Parentheses first E - Exponents (powers and roots) M/D - Multiplication and Division (left to right) A/S - Addition and Subtraction (left to right) Remember: After substitution, you have a regular arithmetic problem! Example 1: With Parentheses Evaluate $3(x + 2)$ when $x = 4$ Solution: Write expression: $3(x + 2)$ Substitute $x = 4$: $3(4 + 2)$ Parentheses first: $3(6)$ Multiply: $18$ Final answer: $18$ Important: $3(x+2)$ is different from $3x+2$! Example 2: Multiple Operations Evaluate $2y^2 - 3y + 5$ when $y = 3$ Solution: Write expression: $2y^2 - 3y + 5$ Substitute $y = 3$: $2(3)^2 - 3(3) + 5$ Exponent first: $2(9) - 3(3) + 5$ Multiply: $18 - 9 + 5$ Left to right: $18 - 9 = 9$, then $9 + 5 = 14$ Final answer: $14$ Example 3: Division Before Addition Evaluate $frac{x + 6}{2}$ when $x = 4$ Solution: Write expression: $frac{x + 6}{2}$ Substitute $x = 4$: $frac{4 + 6}{2}$ Parentheses/numerator first: $frac{10}{2}$ Divide: $5$ Final answer: $5$ Note: $frac{x+6}{2}$ means "the sum of x and 6, divided by 2" Common Mistake: Incorrect: For $3(x+2)$ when $x=4$: $3 times 4 + 2 = 12 + 2 = 14$ Correct: $3(4+2) = 3 times 6 = 18$ Always do parentheses first! The parentheses tell us to add BEFORE multiplying. Application Practice Evaluate $4(a + 3)$ when $a = 2$ Evaluate $2b^2 + 1$ when $b = 3$ Evaluate $frac{x + 8}{4}$ when $x = 12$ Evaluate $3y^2 - 2y$ when $y = 2$ Evaluate $5(2n - 1)$ when $n = 3$ Substitution with Negative Numbers Substituting negative numbers requires special care with signs. Remember the rules for operations with negative numbers. Rules for Negative Numbers: • $-(-a) = +a$ (Two negatives make a positive) • $a - (-b) = a + b$ (Subtracting negative is adding) • $(-a) times (-b) = +ab$ (Negative × negative = positive) • $(-a) times b = -ab$ (Negative × positive = negative) • $a div (-b) = -(a div b)$ (Positive ÷ negative = negative) Example 1: Simple Negative Substitution Evaluate $x + 5$ when $x = -3$ Solution: Write expression: $x + 5$ Substitute $x = -3$: $(-3) + 5$ Add: $2$ Final answer: $2$ Visual: Start at -3 on number line, move 5 right, end at 2 Example 2: Negative Times Positive Evaluate $4y$ when $y = -2$ Solution: Write expression: $4y$ Substitute $y = -2$: $4(-2)$ Multiply: $-8$ Final answer: $-8$ Rule: Positive × negative = negative Example 3: Negative Times Negative Evaluate $-3x$ when $x = -4$ Solution: Write expression: $-3x$ Substitute $x = -4$: $-3(-4)$ Multiply: $12$ Final answer: $12$ Rule: Negative × negative = positive Example 4: Complex Negative Expression Evaluate $2 - x^2$ when $x = -3$ Solution: Write expression: $2 - x^2$ Substitute $x = -3$: $2 - (-3)^2$ Exponent first: $(-3)^2 = 9$ (negative squared is positive) Subtract: $2 - 9 = -7$ Final answer: $-7$ Important: $(-3)^2 = 9$ but $-3^2 = -9$ (without parentheses, only 3 is squared) Technique Practice Evaluate $x + 7$ when $x = -4$ Evaluate $5y$ when $y = -3$ Evaluate $-2n$ when $n = -5$ Evaluate $m^2$ when $m = -6$ Evaluate $8 - k$ when $k = -1$ Substitution in Formulas Formulas are special expressions that represent real-world relationships. Substituting into formulas helps us solve practical problems. Common Formulas: Perimeter of rectangle: $P = 2l + 2w$ Area of rectangle: $A = l times w$ Area of triangle: $A = frac{1}{2}bh$ Distance formula: $d = rt$ (distance = rate × time) Temperature conversion: $F = frac{9}{5}C + 32$ Example 1: Perimeter of Rectangle Find the perimeter of a rectangle with length 8 cm and width 5 cm. Formula: $P = 2l + 2w$ Solution: Write formula: $P = 2l + 2w$ Substitute $l = 8$, $w = 5$: $P = 2(8) + 2(5)$ Multiply: $P = 16 + 10$ Add: $P = 26$ Final answer: $26$ cm Check: Perimeter = 8+5+8+5 = 26 cm ✓ Example 2: Area of Triangle Find the area of a triangle with base 12 m and height 7 m. Formula: $A = frac{1}{2}bh$ Solution: Write formula: $A = frac{1}{2}bh$ Substitute $b = 12$, $h = 7$: $A = frac{1}{2}(12)(7)$ Multiply: $A = frac{1}{2} times 84$ Calculate: $A = 42$ Final answer: $42$ m² Alternative: $A = frac{12 times 7}{2} = frac{84}{2} = 42$ Example 3: Distance Formula If you travel at 60 km/h for 3 hours, how far do you go? Formula: $d = rt$ Solution: Write formula: $d = rt$ Substitute $r = 60$, $t = 3$: $d = 60 times 3$ Multiply: $d = 180$ Final answer: $180$ km Units: km/h × h = km (hours cancel out) Method Practice Find the perimeter of a rectangle with length 10 cm and width 6 cm. Find the area of a triangle with base 8 m and height 5 m. If you travel at 50 km/h for 4 hours, how far do you go? Find the area of a rectangle with length 9 cm and width 4 cm. Convert 20°C to Fahrenheit using $F = frac{9}{5}C + 32$. Multiple Variable Substitution Many expressions and formulas have more than one variable. We substitute values for ALL variables to evaluate the expression. Substituting Multiple Variables: Write the expression with all variables Replace each variable with its given value Use parentheses around each substitution Simplify using order of operations Write final answer with units if applicable Example 1: Two Variables Evaluate $2x + 3y$ when $x = 4$ and $y = 5$ Solution: Write expression: $2x + 3y$ Substitute $x = 4$, $y = 5$: $2(4) + 3(5)$ Multiply: $8 + 15$ Add: $23$ Final answer: $23$ Example 2: Three Variables Evaluate $ab + c$ when $a = 3$, $b = 2$, $c = 7$ Solution: Write expression: $ab + c$ Substitute $a = 3$, $b = 2$, $c = 7$: $(3)(2) + 7$ Multiply: $6 + 7$ Add: $13$ Final answer: $13$ Example 3: With Negative Values Evaluate $p - 2q$ when $p = 10$ and $q = -3$ Solution: Write expression: $p - 2q$ Substitute $p = 10$, $q = -3$: $10 - 2(-3)$ Multiply: $10 - (-6)$ Subtract negative = add: $10 + 6$ Add: $16$ Final answer: $16$ Example 4: Complex Multiple Substitution Evaluate $frac{x + y}{z}$ when $x = 8$, $y = 4$, $z = 3$ Solution: Write expression: $frac{x + y}{z}$ Substitute $x = 8$, $y = 4$, $z = 3$: $frac{8 + 4}{3}$ Add numerator: $frac{12}{3}$ Divide: $4$ Final answer: $4$ Verification Practice Evaluate $3x + 2y$ when $x=4$ and $y=5$ Evaluate $ab - c$ when $a=6$, $b=2$, $c=3$ Evaluate $frac{m+n}{p}$ when $m=10$, $n=6$, $p=4$ Evaluate $2p - 3q$ when $p=7$ and $q=1$ Evaluate $x^2 + y^2$ when $x=5$ and $y=12$ Real-World Formula Applications Formulas help us solve practical problems in science, engineering, finance, and everyday life. The key is to identify which formula to use and what values to substitute. Example 1: Cost Calculation The cost of apples is ₦300 per kilogram. If you buy $n$ kilograms, the cost is $C = 300n$. How much does 5 kg of apples cost? Solution: Formula: $C = 300n$ Substitute $n = 5$: $C = 300 times 5$ Multiply: $C = 1500$ Final answer: ₦1500 If you have ₦900, how many kg can you buy? $900 = 300n$, so $n = 900 div 300 = 3$ kg Example 2: Temperature Conversion To convert Celsius to Fahrenheit: $F = frac{9}{5}C + 32$ Convert 25°C to Fahrenheit. Solution: Formula: $F = frac{9}{5}C + 32$ Substitute $C = 25$: $F = frac{9}{5}(25) + 32$ Multiply: $frac{9}{5} times 25 = 9 times 5 = 45$ Add: $45 + 32 = 77$ Final answer: $77$°F Example 3: Speed Calculation A car travels 240 km in 4 hours. What is its average speed? Formula: $speed = frac{distance}{time}$ or $s = frac{d}{t}$ Solution: Formula: $s = frac{d}{t}$ Substitute $d = 240$, $t = 4$: $s = frac{240}{4}$ Divide: $s = 60$ Final answer: $60$ km/h Example 4: Circle Circumference The circumference of a circle is $C = 2pi r$ where $pi approx 3.14$. Find the circumference of a circle with radius 7 cm. Solution: Formula: $C = 2pi r$ Substitute $pi = 3.14$, $r = 7$: $C = 2 times 3.14 times 7$ Multiply: $2 times 3.14 = 6.28$, then $6.28 times 7 = 43.96$ Final answer: $43.96$ cm Alternative: $2 times 7 times 3.14 = 14 times 3.14 = 43.96$ Application Practice Bananas cost ₦250 per kg. How much do 6 kg cost? (Use $C = 250n$) Convert 30°C to Fahrenheit using $F = frac{9}{5}C + 32$. A cyclist travels 45 km in 3 hours. What is the average speed? Find the perimeter of a square with side length 9 cm. (P = 4s) Find the area of a circle with radius 5 cm. (A = πr², π ≈ 3.14) Checking Your Substitution Checking your work is crucial in substitution. A small mistake can lead to a wrong answer. Here are strategies to verify your substitution. Checking Strategies: Reverse Check: Work backwards from your answer Estimate: Does your answer make sense? Different Method: Solve the problem a different way Substitute Back: Use your answer to check the original Units Check: Do the units make sense? Example 1: Checking with Estimation Evaluate $3x^2 + 5$ when $x = 4$ Your answer: $53$ Check: Original: $3(4)^2 + 5$ $4^2 = 16$, so $3 times 16 = 48$ $48 + 5 = 53$ ✓ Estimate: $4^2=16$, $3times16=48$, $48+5=53$ ✓ Example 2: Checking with Different Values Evaluate $2(x + 3)$ when $x = 5$ Your answer: $16$ Check: Method 1: $2(5+3) = 2times8 = 16$ ✓ Method 2: $2x+6 = 2times5+6 = 10+6=16$ ✓ Method 3: If $x=5$, then $x+3=8$, double it is $16$ ✓ Three different methods all give 16, so answer is correct. Example 3: Checking Units Find area of rectangle: $l=8$ cm, $w=5$ cm Formula: $A = l times w$ Your answer: $40$ Check: Calculation: $8 times 5 = 40$ ✓ Units: cm × cm = cm² ✓ Estimate: $8times5=40$, $10times5=50$, so 40 is reasonable ✓ Visual: 8 by 5 rectangle has 8×5=40 squares ✓ Final answer: $40$ cm² Common Checking Mistakes: Incorrect: Forgetting to check units Correct: Always include units in final answer Incorrect: Not verifying if answer is reasonable Correct: Ask: "Does this number make sense?" Skills Practice Check your answer for $4(x+3)$ when $x=2$ using two methods. Evaluate $3y^2$ when $y=4$ and explain how to check your answer. Find a mistake in this work: For $2(a+5)$ when $a=3$: $2×3+5=6+5=11$ Why is it important to check units in real-world problems? Create a substitution problem and show how to check the answer. Cumulative Exercises Evaluate $5x + 3$ when $x = 4$ Evaluate $2y - 7$ when $y = 6$ Evaluate $frac{a}{4} + 2$ when $a = 12$ Evaluate $b^2 + 5$ when $b = 3$ Evaluate $3(c + 2)$ when $c = 5$ Evaluate $2d^2 - 3$ when $d = 4$ Evaluate $x + 9$ when $x = -4$ Evaluate $5y$ when $y = -2$ Evaluate $-3n$ when $n = -5$ Evaluate $m^2$ when $m = -7$ Find perimeter of rectangle: length=9 cm, width=6 cm (P=2l+2w) Find area of triangle: base=10 m, height=8 m (A=½bh) Evaluate $2x + 3y$ when $x=4$, $y=5$ Evaluate $ab - c$ when $a=7$, $b=3$, $c=4$ Convert 15°C to Fahrenheit (F=⁹⁄₅C+32) Bananas cost ₦250 per kg. How much for 8 kg? (C=250n) Car travels 180 km in 3 hours. Find speed. (s=d/t) Evaluate $frac{p+q}{r}$ when $p=15$, $q=9$, $r=6$ Evaluate $4x^2 - 2x$ when $x=3$ Create a real-world substitution problem and solve it. Show/Hide Answers Exercise 1: $5(4)+3=20+3=23$ Exercise 2: $2(6)-7=12-7=5$ Exercise 3: $frac{12}{4}+2=3+2=5$ Exercise 4: $3^2+5=9+5=14$ Exercise 5: $3(5+2)=3×7=21$ Exercise 6: $2(4)^2-3=2×16-3=32-3=29$ Exercise 7: $(-4)+9=5$ Exercise 8: $5(-2)=-10$ Exercise 9: $-3(-5)=15$ Exercise 10: $(-7)^2=49$ Exercise 11: $P=2(9)+2(6)=18+12=30$ cm Exercise 12: $A=frac{1}{2}(10)(8)=frac{1}{2}×80=40$ m² Exercise 13: $2(4)+3(5)=8+15=23$ Exercise 14: $(7)(3)-4=21-4=17$ Exercise 15: $F=frac{9}{5}(15)+32=27+32=59$°F Exercise 16: $C=250×8=2000$ ₦ Exercise 17: $s=frac{180}{3}=60$ km/h Exercise 18: $frac{15+9}{6}=frac{24}{6}=4$ Exercise 19: $4(3)^2-2(3)=4×9-6=36-6=30$ Exercise 20: Sample: Pizza costs ₦1500 each. For n pizzas, cost C=1500n. 3 pizzas cost 1500×3=₦4500 Conclusion/Recap Excellent work! You've now mastered formula substitution - replacing variables with values in expressions and formulas. This skill connects abstract algebra with practical calculation and is essential for all future mathematics. Key Concepts to Remember: 1. Variables: Letters that represent unknown or changing values 2. Substitution: Replace each variable with its given value 3. Order of Operations: PEMDAS/BODMAS must be followed 4. Negative Numbers: Careful with signs when substituting negatives 5. Formulas: Special expressions for real-world relationships 6. Multiple Variables: Substitute all variables with their values 7. Checking: Always verify your answer makes sense Common Mistakes to Avoid: • Forgetting parentheses when substituting • Incorrect order of operations • Mishandling negative signs • Confusing $3x$ (3 times x) with "thirty-something" • Forgetting to include units in final answers • Not checking if answer is reasonable Formula substitution is used in science, engineering, finance, and everyday life. Every time you calculate an area, convert temperatures, determine costs, or compute speed, you're using substitution. Keep practicing by looking for formulas in your textbooks, in recipes, on maps, and in instructions. Clip It! Share your ANSWER in the Chat. Indicate TITLE e.g Linear Equation 1. .....2. e.t.c