Question
(. One system below is set up for substitution and the other is set up for elimination. EXPLAIN how \( \begin{array}{ll}\text { you can tell which strategy each system below is set up for. You do not need to solve the system. } \\ \begin{array}{ll}\text { a. } \\ x+y=3 & \text { b. } y=3 x+4 \\ x-y=1 & \\ 2 x+y=8\end{array}\end{array} \)
Ask by Gross Summers. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
**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.
Solution
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.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
In the first system, the equation \(x + y = 3\) is already in a form that allows you to easily isolate one variable, making substitution the natural choice. You can simply solve for \(y\) as \(y = 3 - x\) and substitute it into the second equation, \(x - y = 1\). On the other hand, the second system has the equation \(y = 3x + 4\), which conveniently expresses \(y\) in terms of \(x\}; this leads to a straightforward application of substitution as well. However, with the equation \(2x + y = 8\), you could also eliminate \(y\) by substituting \(3x + 4\) into it, showing that this system can lead to either method but is primarily suited for substitution.
