Solve Linear System with Parameters and Count Solutions

Question:

Let the system of equations

$$\begin{cases} x + 5y - z = 1 \\[1em] 4x + 3y - 3z = 7 \\[1em] 24x + y + \lambda z = \mu,\quad \lambda, \mu \in \mathbb{R} \end{cases}$$

have infinitely many solutions.
Then, the number of solutions of this system, if $x, y, z$ are integers and satisfy

$$7\le x+y+z\le 77,$$

is:

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Question Image

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Short Solution

Idea:

1. For infinitely many solutions, the third equation must be a linear combination of the first two.

2. Find $\lambda$ and $\mu$.

3. Reduce to two equations in $x, y, z$, find the relation among variables.

4. Put $x + y + z = t$, derive bounds $7\le t\le 77$

5. Count integer solutions.

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Image Solution

Step 1: Check dependence
We have:

$$E_1:x+5y-z=1\mathrm{<br>}$$$$E_2: 4x + 3y - 3z = 7$$

Let $E_3: 24x + y + \lambda z = \mu$
For infinite solutions:

$$E_3=p{E}_1+q{E}_2$$

Matching coefficients:

$$\begin{gathered} 24=p(1)+q(4) \\ 1=5p+3q \\ \lambda =-p-3q \\ \mu =p(1)+q(7) \end{gathered}$$

Solve:

• From first: $p + 4q = 24$
  

• From second: $5p + 3q = 1$
  

Multiply first by 5:

$$5p+20q=120$$

Subtract:

$$17q=119\Rightarrow q=7,p=-4.$$

Then:

$$\lambda =-(-4)-21=4-21=-17,\mu =-4+49=45.$$

So $E_3$ is consistent.

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Step 2: Solve the pair
Take $E_1$ and $E_2$:

$$\begin{array}{cccc}& x+5y-z=1& & \text{(1)}\end{array}$$$$\begin{array}{cccc}& 4x+3y-3z=7& & \text{(2)}\end{array}$$

Multiply (1) by 4:

$$4x+20y-4z=4$$

Subtract from (2):

$$-17y + z = 3 \Rightarrow z = 17y + 3.$$From (1):

$$x+5y-(17y+3)=1\Rightarrow x-12y-3=1\Rightarrow x=12y+4.$$

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Step 3: Sum constraint

$$x+y+z=12y+4+y+17y+3=30y+7.$$

So:

$$t=30y+7,7\le t\le 77.$$

Subtract 7:

$$0\le 30y\le 70\Rightarrow 0\le y\le \frac{70}{30}=\frac{7}{3}\mathrm{.}$$

So $y=0,1,2$

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Step 4: Solutions

$y$	$x$	$z$	$x+y+z$
0	4	3	7
1	16	20	37
2	28	37	67

All lie between $7$ and $77$.

✅ Number of solutions = 3

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Conclusion & Video Solution

The system has exactly 3 integer solutions satisfying the given condition.

$$\boxed{3}$$

👉 Watch the detailed video explanation here: