Age Word Problem Using Substitution Method

 

❓ Concept Question

How can we solve age-related word problems using a pair of linear equations by the Substitution Method?

---

🖼️ Concept Image

---

✍️ Short Concept

This concept is based on:

👉 Pair of Linear Equations in Two Variables

👉 Formation of equations from word problems

👉 Substitution Method

---

🔷 Step 1 — Assume Variables

Let:

• $x$ = Aftab's present age (in years)
• $y$ = Daughter's present age (in years)

The first step in any age problem is to represent the unknown ages using variables.

---

🔷 Step 2 — Form the Equations

Condition 1: Seven years ago

Aftab says,

"Seven years ago, I was seven times as old as you were then."

So,

$$x-7=7(y-7)$$

Expanding,

$$x-7y=-42$$

Condition 2: Three years from now

Aftab says,

"Three years from now, I shall be three times as old as you will be."

So,

$$x+3=3(y+3)$$

Simplifying,

$$x=3y+6$$

---

🔷 Step 3 — Apply the Substitution Method

Substitute

$$x=3y+6$$

into

$$x-7y=-42$$
$$(3y+6)-7y=-42$$
$$-4y+6=-42$$
$$-4y=-48$$
$$y=12$$

Now substitute $y=12$ into

$$x=3y+6$$
$$x=3(12)+6=42$$

Thus,

• Aftab's present age = 42 years
• Daughter's present age = 12 years

---

🔷 Step 4 — Graphical Representation

The two equations are:

$$x-7y=-42$$

and

$$x-3y=6$$

When these two lines are drawn on a graph, they intersect at the point:

$$(42,\,12)$$

This intersection point gives the solution of the pair of linear equations.

---

🔷 Step 5 — Verify the Answer

Seven years ago:

Aftab's age $=42-7=35$

Daughter's age $=12-7=5$

$$35=7\times5$$

✔ Condition satisfied.

Three years from now:

Aftab's age $=42+3=45$

Daughter's age $=12+3=15$

$$45=3\times15$$

✔ Condition satisfied.

---

🚨 Common Mistakes

❌ Using present ages instead of ages "seven years ago" or "three years from now"

❌ Forgetting to apply brackets:

$$7(y-7)\neq 7y-7$$

It should be:

$$7(y-7)=7y-49$$

❌ Making sign errors while simplifying the equations

---

✅ Final Takeaway

To solve age word problems:

✔ Assume variables for present ages.

✔ Convert each statement into a linear equation.

✔ Solve the equations using the substitution method.

✔ Verify the answer using the original conditions.

---

⭐ Class 10 Insight

In age-related problems, carefully read phrases like:

• "Years ago" → subtract years.
• "Years from now" → add years.
• "Times as old" → multiply accordingly.

Converting the language into equations correctly is the key to solving these questions.

---

📚 Related Topics

• Find Zeroes and Verify Relationship Between Coefficients
• Find Quadratic Polynomial from Sum and Product of Zeroes
• Quadratic Polynomial with Irrational Zeroes Explained