(. One system below is set up for substitution and the other is set up for elimination. EXPLAIN how $$ \begin{array}{ll} ext { you can tell which strategy each system below is set up for. You do not need to solve the system. } \ \begin{array}{ll} ext { a. } \ x+y=3 & ext { b. } y=3 x+4 \ x-y=1 & \ 2 x+y=8\end{array}\end{array} $$

Question

(. One system below is set up for substitution and the other is set up for elimination. EXPLAIN how $$ \begin{array}{ll}	ext { you can tell which strategy each system below is set up for. You do not need to solve the system. } \ \begin{array}{ll}	ext { a. } \ x+y=3 & 	ext { b. } y=3 x+4 \ x-y=1 & \ 2 x+y=8\end{array}\end{array} $$

UpStudy AI · Accepted Answer

To determine whether each system is set up for **substitution** or **elimination**, observe the form in which the equations are presented:

### a. 
$$
\begin{cases}
x + y = 3 \
x - y = 1
\end{cases}
$$
**Strategy: Elimination**

**Explanation:**
- **Standard Form:** Both equations are written in the standard linear form $$ Ax + By = C $$.
- **Coefficient Alignment:** The coefficients of $$ y $$ are $$ +1 $$ and $$ -1 $$, which are opposites.
- **Ease of Elimination:** Adding or subtracting the equations will eliminate one variable directly. For instance, adding both equations:
  $$
  (x + y) + (x - y) = 3 + 1 \implies 2x = 4 \implies x = 2
  $$
  This straightforward elimination of $$ y $$ indicates that the system is set up for the **elimination** method.

### b. 
$$
\begin{cases}
y = 3x + 4 \
2x + y = 8
\end{cases}
$$
**Strategy: Substitution**

**Explanation:**
- **Isolation of a Variable:** The first equation is already solved for $$ y $$ ($$ y = 3x + 4 $$).
- **Direct Substitution:** This setup allows you to substitute $$ y $$ from the first equation into the second equation seamlessly:
  $$
  2x + (3x + 4) = 8
  $$
- **Simplifies to One Variable:** Substituting reduces the system to a single equation with one variable, facilitating the **substitution** method.

### Summary
- **System a** is arranged in a way that allows for the **elimination** of a variable by adding or subtracting the equations.
- **System b** is arranged to easily **substitute** one equation into the other by having one variable already isolated.

By examining the arrangement of each system—whether a variable is isolated for easy substitution or coefficients are aligned for canceling—a system can be identified as set up for substitution or elimination.
**System a** is set up for **elimination** because both equations are in standard form with opposite coefficients for one variable, making it easy to eliminate that variable by adding or subtracting the equations. **System b** is set up for **substitution** because one equation is already solved for a variable, allowing you to substitute directly into the other equation. So, **System a** uses elimination, and **System b** uses substitution.

Solución de tutoría real

Responder

Solución

Mind Expander

preguntas relacionadas

Latest Algebra Questions