SAT-Math-19: Systems with Infinite Solutions

1. The Geometric & Algebraic Meaning

When we solve a system of equations, we are looking for the point where two lines cross. However, what happens if the two equations describe the exact same line?

If you graph them, one line sits directly on top of the other. Because they touch at every single coordinate along their infinite path, we say the system has infinitely many solutions.

The Proportionality Rule

For a system to have infinite solutions, the standard form equations must be completely equivalent. If you look at the coefficients:

$$\begin{cases} A_1x + B_1y = C_1 \\ A_2x + B_2y = C_2 \end{cases}$$

The ratios of the coefficients and the constants must all be identical:

$$\frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2}$$

🇺🇿 Uzbek Explanation:

Cheksiz ko'p yechimli sistemalar: Agar sistemadagi ikkala tenglama ham aslida bitta bir xil chiziqni ifodalasa, ularning kesishish nuqtalari cheksiz ko'p bo'ladi. Buni topshiriqlarda tez aniqlash siri: ikkinchi tenglamadagi barcha sonlar birinchisidagilardan bir xil marta katta yoki kichik bo'ladi (proporsional bo'ladi).

2. High-Yield SAT Practice Question

Question: In the system of equations below, $k$ and $p$ are constants. If the system has infinitely many solutions, what is the value of $k + p$?

$$\begin{cases} 3x + 5y = 12 \\ kx + 15y = p \end{cases}$$

A) 9

B) 21

C) 36

D) 45

Step-by-Step Explanation:

1. Find the scaling factor: Look at the variable that has numbers provided in both equations. Here, it’s the $y$-term.

   • First equation has $+5y$.

   • Second equation has $+15y$.

2. Ask yourself: What do I multiply $5$ by to get $15$?

   $$5 \times 3 = 15$$

   The scaling factor is exactly 3.

3. Since the system has infinitely many solutions, the entire first equation must be multiplied by 3 to create the second equation.

   $$3 \cdot (3x + 5y = 12) \implies 9x + 15y = 36$$

4. Match the coefficients to find our missing constants:

   • $k = 9$

   • $p = 36$

5. Answer the ultimate request: The question asks for the value of $k + p$.

   $$k + p = 9 + 36 = 45$$

Correct Answer: D

Summary

• Infinite Solutions = Identical Lines: Both equations share the exact same graph, same slope, and same $y$-intercept.

• The Scale Factor Hack: Find the multiplier between the known coefficients and apply it across the entire equation to solve for unknown constants instantly.

• Always check if the question asks for a single constant ($k$) or a combination ($k+p$).

You are executing like a master, my bro! Ready to see the exact opposite scenario in SAT-Math-20: Systems with No Solution (Parallel Lines)?