Substitution Method Practice Questions

 

❓ Concept Question

How do we solve a pair of linear equations using the Substitution Method?

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🖼️ Concept Image

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✍️ Short Concept

This concept is based on:

👉 Pair of Linear Equations in Two Variables

👉 Substitution Method

👉 Consistent and Inconsistent Systems

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🔷 Step 1 — What is the Substitution Method?

For two equations:

$$a_1x+b_1y=c_1$$
$$a_2x+b_2y=c_2$$

We:

1. Express one variable in terms of the other.
2. Substitute into the second equation.
3. Find one variable.
4. Substitute back to find the other variable.

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🔷 Example (i)

Solve:

$$x+y=14$$
$$x-y=4$$

From first equation:

$$x=14-y$$

Substitute into second equation:

$$14-y-y=4$$
$$14-2y=4$$
$$y=5$$

Substituting back:

$$x=14-5=9$$

✅ Solution

$$x=9,\quad y=5$$

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🔷 Example (ii)

Solve:

$$s-t=3$$
$$\frac{s}{3}+\frac{t}{2}=6$$

From first equation:

$$s=3+t$$

Substitute:

$$\frac{3+t}{3}+\frac{t}{2}=6$$    

Multiply by 6:

$$2(3+t)+3t=36$$
$$6+5t=36$$
$$t=6$$
$$s=3+6=9$$

✅ Solution

$$s=9,\quad t=6$$

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🔷 Example (iii)

Solve:

$$3x-y=3$$
$$9x-3y=9$$

Multiply first equation by 3:

$$9x-3y=9$$

This is exactly the second equation.

✅ Result

Both equations represent the same line.

Therefore:

$$\boxed{\text{Infinitely many solutions}}$$

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🔷 Example (iv)

Solve:

$$0.2x+0.3y=1.3$$
$$0.4x+0.5y=2.3$$

Multiply both equations by 10:

$$2x+3y=13$$
$$4x+5y=23$$

From first equation:

$$2x=13-3y$$

Substitute into second equation:

$$2(13-3y)+5y=23$$
$$26-y=23$$
$$y=3$$
$$x=2$$

✅ Solution

$$x=2,\quad y=3$$

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🔷 Example (v)

Solve:

$$\sqrt2\,x+\sqrt3\,y=0$$
$$\sqrt3\,x-\sqrt8\,y=0$$

From first equation:

$$x=-\frac{\sqrt3}{\sqrt2}y$$

Substituting into second equation gives:

$$y=0$$

Hence:

$$x=0$$

✅ Solution

$$x=0,\quad y=0$$

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🔷 Example (vi)

Solve:

$$\frac{3x}{2}-\frac{5y}{3}=-2$$
$$\frac{x}{3}+\frac{y}{2}=\frac{13}{6}$$

Multiply both equations by 6:

$$9x-10y=-12$$
$$2x+3y=13$$

From second equation:

$$x=\frac{13-3y}{2}$$

Substituting into first equation:

$$9\left(\frac{13-3y}{2}\right)-10y=-12$$

Solving:

$$y=3$$
$$x=2$$

✅ Solution

$$x=2,\quad y=3$$

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🚨 Common Mistakes

❌ Forgetting to substitute the complete expression.

❌ Sign mistakes while simplifying.

❌ Not clearing fractions before solving.

❌ Missing the special case where one equation is a multiple of the other.

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✅ Final Takeaway

Using the substitution method:

Part	Solution
(i)	$x=9,y=5$
(ii)	$s=9,\ t=6$
(iii)	Infinitely many solutions
(iv)	$x=2,y=3$
(v)	$x=0,y=0$
(vi)	$x=2,y=3$

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⭐ Class 10 Insight

Before solving:

✔ Check whether one equation is already easy to write in terms of a variable.

✔ Remove decimals and fractions first whenever possible.

✔ If one equation becomes identical to the other, the system has infinitely many solutions.

The Substitution Method is one of the most important methods for solving CBSE Class 10 linear equations.

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📚 Related Topics

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