Substitution

What Is Substitution?

Substitution is one of the most fundamental skills in algebra. It's the process of replacing a variable with its actual value. Think of it like this: if a variable is a placeholder or a blank space, substitution is filling in that blank with a number.

You use substitution constantly in real life, even if you don't realize it. When you follow a recipe that says "bake for 15 minutes per pound," you're substituting the weight of your food into a formula. When you calculate your earnings (hours worked times hourly rate), you're substituting your specific hours and rate into the expression.

In algebra, letters like \(x\), \(y\), and \(a\) represent unknown or changing values. Once you know what those values are, substitute them in to find the answer.

Substituting Into Expressions

An expression is a mathematical phrase that can contain numbers, variables, and operations. When you substitute into an expression, you're replacing the variable(s) with specific number(s) and then calculating the result.

Example 1: Evaluate \(3x + 5\) when \(x = 4\).

Take the expression \(3x + 5\) and replace \(x\) with \(4\):

$$3x + 5$$ $$3(4) + 5$$ $$12 + 5$$ $$17$$

Answer: \(17\)

Notice the parentheses around the substituted value. They aren't strictly required here, but they prevent mistakes once negative numbers or more complex expressions enter the picture.

Example 2: Evaluate \(7y - 2\) when \(y = 3\).

Replace \(y\) with \(3\):

$$7y - 2$$ $$7(3) - 2$$ $$21 - 2$$ $$19$$

Answer: \(19\)

Example 3: Evaluate \(2a + 8\) when \(a = 0\).

Don't forget that zero is a valid value to substitute:

$$2a + 8$$ $$2(0) + 8$$ $$0 + 8$$ $$8$$

Answer: \(8\)

Even though \(a\) equals zero and the \(2a\) term disappears, the constant \(8\) remains.

Example 4: Evaluate \(x^2 + 4x\) when \(x = 5\).

This one has an exponent. Remember to apply the exponent to the substituted value:

$$x^2 + 4x$$ $$(5)^2 + 4(5)$$ $$25 + 20$$ $$45$$

Answer: \(45\)

The order of operations matters here. Square the 5 first to get 25, then multiply 4 times 5 to get 20, then add.

Example 5: Evaluate \(\frac{n}{2} + 7\) when \(n = 10\).

Substitution works with fractions too:

$$\frac{n}{2} + 7$$ $$\frac{10}{2} + 7$$ $$5 + 7$$ $$12$$

Answer: \(12\)

Substituting Into Equations

An equation has an equals sign, which means there's a left side and a right side that should be equal. You can substitute into equations to check if a particular value makes the equation true, or to find the value of one variable when you know another.

Example 6: Is \(x = 6\) a solution to the equation \(2x + 3 = 15\)?

To check, substitute \(6\) for \(x\) and see if both sides are equal:

$$2x + 3 = 15$$ $$2(6) + 3 = 15$$ $$12 + 3 = 15$$ $$15 = 15$$ ✓

Yes, it checks out! When \(x = 6\), both sides equal 15, so \(x = 6\) is indeed a solution.

Example 7: Is \(y = 3\) a solution to the equation \(5y - 4 = 12\)?

Substitute and check:

$$5y - 4 = 12$$ $$5(3) - 4 = 12$$ $$15 - 4 = 12$$ $$11 = 12$$ ✗

No, this doesn't work. When \(y = 3\), the left side equals 11, not 12. So \(y = 3\) is not a solution.

Example 8: If \(3a + 7 = b\) and \(a = 2\), what is \(b\)?

Substitute the known value and solve for the unknown:

$$3a + 7 = b$$ $$3(2) + 7 = b$$ $$6 + 7 = b$$ $$13 = b$$

Answer: \(b = 13\)

Example 9: If \(x + y = 12\) and \(x = 5\), what is \(y\)?

Substitute the known value:

$$x + y = 12$$ $$5 + y = 12$$

Now solve for \(y\):

$$y = 12 - 5$$ $$y = 7$$

Answer: \(y = 7\)

Working With Formulas

Formulas are equations that express relationships between quantities. Substitution is the bridge between a formula and a specific numerical answer.

Example 10: The formula for the area of a rectangle is \(A = lw\), where \(l\) is length and \(w\) is width. Find the area when \(l = 8\) and \(w = 5\).

Substitute both values:

$$A = lw$$ $$A = (8)(5)$$ $$A = 40$$

Answer: The area is 40 square units.

Example 11: The formula for the perimeter of a rectangle is \(P = 2l + 2w\). Find \(P\) when \(l = 12\) and \(w = 7\).

$$P = 2l + 2w$$ $$P = 2(12) + 2(7)$$ $$P = 24 + 14$$ $$P = 38$$

Answer: The perimeter is 38 units.

Example 12: The distance formula is \(d = rt\), where \(d\) is distance, \(r\) is rate (speed), and \(t\) is time. If you travel at 60 miles per hour for 3 hours, how far do you go?

Substitute \(r = 60\) and \(t = 3\):

$$d = rt$$ $$d = (60)(3)$$ $$d = 180$$

Answer: You travel 180 miles.

Example 13: The formula to convert Celsius to Fahrenheit is \(F = \frac{9}{5}C + 32\). What is 25°C in Fahrenheit?

Substitute \(C = 25\):

$$F = \frac{9}{5}C + 32$$ $$F = \frac{9}{5}(25) + 32$$ $$F = \frac{225}{5} + 32$$ $$F = 45 + 32$$ $$F = 77$$

Answer: 25°C equals 77°F.

This is a practical example of substitution you might actually use if you're traveling to a country that uses different temperature units.

Multiple Variables

When an expression or equation has multiple variables, substitute all the known values at once.

Example 14: Evaluate \(3x + 2y\) when \(x = 4\) and \(y = 5\).

Replace both variables:

$$3x + 2y$$ $$3(4) + 2(5)$$ $$12 + 10$$ $$22$$

Answer: \(22\)

Example 15: Evaluate \(a^2 + b^2\) when \(a = 3\) and \(b = 4\).

$$a^2 + b^2$$ $$(3)^2 + (4)^2$$ $$9 + 16$$ $$25$$

Answer: \(25\)

This is actually part of the Pythagorean theorem. If you have a right triangle with legs of length 3 and 4, the hypotenuse would be \(\sqrt{25} = 5\).

Example 16: Evaluate \(xy + z\) when \(x = 2\), \(y = 6\), and \(z = 3\).

$$xy + z$$ $$(2)(6) + 3$$ $$12 + 3$$ $$15$$

Answer: \(15\)

Example 17: Evaluate \(\frac{a + b}{2}\) when \(a = 8\) and \(b = 12\).

This expression finds the average (mean) of two numbers:

$$\frac{a + b}{2}$$ $$\frac{8 + 12}{2}$$ $$\frac{20}{2}$$ $$10$$

Answer: \(10\)

Substituting Negative Values

Substituting negative numbers requires extra care. Always use parentheses around negative values to avoid sign errors.

Example 18: Evaluate \(5x + 3\) when \(x = -2\).

Use parentheses around the \(-2\):

$$5x + 3$$ $$5(-2) + 3$$ $$-10 + 3$$ $$-7$$

Answer: \(-7\)

Example 19: Evaluate \(x^2 - 4\) when \(x = -3\).

This is where parentheses really matter:

$$x^2 - 4$$ $$(-3)^2 - 4$$ $$9 - 4$$ $$5$$

Answer: \(5\)

Notice that \((-3)^2 = 9\), not \(-9\). When you square a negative number, you get a positive result. The parentheses remind you to square the entire negative number.

Example 20: Evaluate \(2y - 5\) when \(y = -4\).

$$2y - 5$$ $$2(-4) - 5$$ $$-8 - 5$$ $$-13$$

Answer: \(-13\)

Example 21: Evaluate \(a - b\) when \(a = 6\) and \(b = -2\).

Be careful with the double negative here:

$$a - b$$ $$6 - (-2)$$ $$6 + 2$$ $$8$$

Answer: \(8\)

Subtracting a negative is the same as adding a positive. This is one of the most common places students make mistakes.

Example 22: Evaluate \(-3x + 7\) when \(x = -5\).

$$-3x + 7$$ $$-3(-5) + 7$$ $$15 + 7$$ $$22$$

Answer: \(22\)

When you multiply two negatives, you get a positive, so \(-3 \times -5 = 15\).

Quick check: evaluate \(x^2 + x\) when \(x = -4\). Show answer\((-4)^2 + (-4) = 16 + (-4) = 12\). Remember to square the whole negative number, then add the negative value.

Try These Problems

Work through these on your own, then check your answers.

1. Evaluate \(4x + 1\) when \(x = 3\) Show answer13 — \(4(3) + 1 = 12 + 1 = 13\)

2. Evaluate \(10 - 2y\) when \(y = 4\) Show answer2 — \(10 - 2(4) = 10 - 8 = 2\)

3. Evaluate \(n^2 + 5\) when \(n = 6\) Show answer41 — \((6)^2 + 5 = 36 + 5 = 41\)

4. Evaluate \(\frac{x}{3} + 2\) when \(x = 9\) Show answer5 — \(\frac{9}{3} + 2 = 3 + 2 = 5\)

5. Is \(t = 7\) a solution to \(3t - 5 = 16\)? Show answerYes — \(3(7) - 5 = 21 - 5 = 16\) ✓

6. If \(2a + 3 = b\) and \(a = 4\), find \(b\) Show answer\(b = 11\) — \(2(4) + 3 = 8 + 3 = 11\)

7. Evaluate \(5m + 3n\) when \(m = 2\) and \(n = 4\) Show answer22 — \(5(2) + 3(4) = 10 + 12 = 22\)

8. Evaluate \(x^2 - y^2\) when \(x = 5\) and \(y = 3\) Show answer16 — \((5)^2 - (3)^2 = 25 - 9 = 16\)

9. Evaluate \(3x - 7\) when \(x = -2\) Show answer-13 — \(3(-2) - 7 = -6 - 7 = -13\)

10. Evaluate \(a^2\) when \(a = -4\) Show answer16 — \((-4)^2 = 16\), not \(-16\)! Squaring a negative always gives a positive result.

Why Substitution Matters

Substitution shows up constantly throughout algebra and beyond. Every time you plug numbers into a formula — area of a room, compound interest, a scientific equation — that's substitution. It's also the engine inside other techniques: solving systems of equations by substitution, evaluating functions at an input value, and verifying that a solution actually works by plugging it back into the original equation.

A few traps come up over and over. Always put parentheses around negative substituted values: \(x = -3\) into \(2x\) is \(2(-3) = -6\), and the parentheses keep the sign straight. Squaring a negative gives a positive: \((-5)^2 = 25\), not \(-25\) — but written without parentheses, \(-5^2\) means \(-(5^2) = -25\). Follow the order of operations: \(3x^2\) with \(x = 4\) is \(3(4)^2 = 3(16) = 48\), not \((3 \cdot 4)^2 = 144\). Don't drop constant terms when the variable term hits zero: \(5x + 9\) at \(x = 0\) is \(9\), not \(0\). And subtracting a negative is adding: \(a - (-3) = a + 3\).

Master substitution now and it'll feel automatic everywhere it comes up later.