SAT-Math-16: Systems of Linear Equations: Solving by Substitution

1. What is a System of Equations?

A system of equations is a set of two or more equations containing the same variables. The solution to a system is the specific coordinate point $(x, y)$ where the two lines intersect on a graph. This point satisfies both equations simultaneously.

There are two main algebraic methods to solve a system: Substitution and Elimination. Today, we focus on Substitution.

2. The Substitution Strategy

The substitution method is cleanest when one of the equations already has a variable completely isolated (e.g., $y = \dots$ or $x = \dots$), or when a variable has a coefficient of $1$.

The 3-Step Execution Blueprint

1. Isolate: Find the simplest equation and isolate one variable (if not already done).

2. Substitute: Plug that entire expression into the other equation in place of the isolated variable. This collapses the system into a single equation with only one variable.

3. Solve & Back-solve: Solve for that single variable, then plug the numerical result back into your original expression to find the remaining variable.

🇺🇿 Uzbek Explanation:

O'rin qoyish usuli (Substitution Method): Bu usul chiziqli tenglamalar sistemasini yechishda ishlatiladi. Agar tenglamalarning birida biron bir harf yolg'iz turgan bo'lsa (masalan, $y = 2x + 3$), ushbu ifodani ikkinchi tenglamadagi $y$ ning o'rniga olib borib qo'yamiz. Natijada bitta noma'lumli oddiy tenglama hosil bo'ladi.

3. High-Yield SAT Practice Question

Question:

$$\begin{cases} y = 2x - 3 \\ 3x + 2y = 22 \end{cases}$$

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

A) $4$

B) $5$

C) $9$

D) $13$

Step-by-Step Explanation:

1. Identify the shortcut: The first equation already has $y$ isolated: $y = 2x - 3$. This screams substitution!

2. Substitute into the second equation: Substitute the chunk $(2x - 3)$ wherever you see $y$ in the second equation:

   $$3x + 2(2x - 3) = 22$$

3. Solve for $x$:

   • Distribute the $2$:

     $$3x + 4x - 6 = 22$$

   • Combine like terms:

     $$7x - 6 = 22$$

   • Add $6$:

     $$7x = 28$$

   • Divide by $7$:

     $$x = 4$$

4. Back-solve for $y$: Plug $x = 4$ back into our isolated equation:

   $$y = 2(4) - 3 \implies y = 8 - 3 \implies y = 5$$

5. Answer the actual question: The prompt asks for the value of $x + y$, not just $x$ or $y$.

   $$x + y = 4 + 5 = 9$$

Correct Answer: C

Summary

• Choose substitution when an equation has a variable with a coefficient of $1$ or is already isolated.

• Use parentheses when substituting an expression to preserve proper distribution of coefficients.

• Read the final line: Always make sure you calculate exactly what the prompt asks for (e.g., $x+y$, $2x$, or $y-x$).

Fantastic workflow, my bro! Are you ready to cross over to the next tool: SAT-Math-17: Systems of Linear Equations: Solving by Elimination?