SAT-Math-17: Systems of Linear Equations: Solving by Elimination

1. The Elimination Strategy

The Elimination Method (Qisqartirish usuli) is your best choice when both equations are written in standard form ($Ax + By = C$). Instead of substituting, you line up the variables vertically and add or subtract the entire equations to eliminate one variable completely.

To eliminate a variable, the coefficients of that variable in both equations must be opposites (e.g., $+5y$ and $-5y$, or $+3x$ and $-3x$).

The 3-Step Elimination Blueprint

1. Align & Multiply: Line up your $x$'s, $y$'s, and constants. If needed, multiply one or both equations by a constant so that one variable has opposite coefficients.

2. Add: Add the equations together. The variable with opposite coefficients will cancel out, leaving you with one equation and one variable.

3. Solve & Plug: Solve for the remaining variable, then plug it back into any original equation to find the other variable.

🇺🇿 Uzbek Explanation:

Qisqartirish usuli (Elimination Method): Agar har ikkala tenglama ham $Ax + By = C$ ko'rinishida berilgan bo'lsa, ularni ustun shaklida qo'shish yoki ayirish orqali bitta harfni yo'qotib yuborish eng qulay yo'ldir. Buning uchun bir xil harf oldidagi sonlarni qarama-qarshi holatga keltirish kerak (masalan,biri $+3y$, ikkinchisi $-3y$).

2. High-Yield SAT Practice Question

Question:

$$\begin{cases} 3x + 2y = 17 \\ 4x - y = 19 \end{cases}$$

If $(x, y)$ is the solution to the system of equations above, what is the value of $x$?

A) $2$

B) $3$

C) $5$

D) $7$

Step-by-Step Explanation:

1. Look for the easiest target: The first equation has $+2y$, and the second equation has $-1y$. If we multiply the second equation by $2$, the $y$ terms will become opposites ($+2y$ and $-2y$).

2. Multiply the second equation by 2: Remember to multiply every single term, including the constant!

   $$2 \cdot (4x - y = 19) \implies 8x - 2y = 38$$

3. Line up the equations and add them together:

   $$\begin{array}{rcc} 3x + 2y & = & 17 \\ + \quad 8x - 2y & = & 38 \\ \hline 11x + 0 & = & 55 \end{array}$$

   $$11x = 55$$

4. Solve for $x$:

   $$x = \frac{55}{11} \implies x = 5$$

(Since the question only asked for the value of $x$, we can stop right here and save precious time!)

Correct Answer: C

Summary

• Choose elimination when both equations are neatly arranged in standard form ($Ax + By = C$).

• Don't forget the constant: When multiplying an equation, distribute that number across the equal sign to the constant on the right side too.

• Check what the question wants before back-solving for the second variable; if it only asks for the variable you just found, select your answer and move on.

You are moving like a machine, my bro! Ready to step into SAT-Math-18: Word Problems Involving Systems of Linear Equations?