Serina wants to solve the following system of equations in the most efficient way.

[tex]$\begin{array}{r}
2 x+3 y=18 \
x+7 y=31
\end{array}$[/tex]

She plans to solve for [tex]$x$[/tex] in the first equation as her first step since both 2 and 3 can be divided into 18. Why is Serina mistaken?
A. Serina should have solved for [tex]$x$[/tex] in the second equation because it has a coefficient of 1.
B. Serina should have solved for [tex]$y$[/tex] in the first equation because dividing by 3 instead of by 2 would give a smaller number in the solution.
C. Serina should have solved for [tex]$y$[/tex] in the second equation because it has the largest coefficient.
D. Serina should have solved for [tex]$y$[/tex] in the first equation because the division step will be easier since 18 is divisible by 3.

Serina's mistake lies in her choice to solve for x in the first equation instead of the second, where x has a coefficient of 1. This makes isolating x in the second equation more efficient. Thus, she should opt to solve for x in the second equation as it simplifies the calculations.
;

Answered by Anonymous | 2025-07-04

Analyze the system of equations and identify the coefficients of x and y in both equations.
Evaluate the ease of isolating each variable in each equation, looking for coefficients of 1 or divisibility properties.
Compare the complexity of the resulting expressions after isolating each variable.
Determine that solving for x in the second equation is the most efficient because it has a coefficient of 1, avoiding fractions in the initial step. Therefore, Serina should have solved for x in the second equation because it has a coefficient of 1.

Explanation

Analyze Serina's Plan Serina's plan is to solve for x in the first equation because 2 and 3 can be divided into 18. We need to determine why this is not the most efficient way to solve the system of equations. The system of equations is:

First Equation 2 x + 3 y = 18

Second Equation x + 7 y = 31

Analyze Coefficients Let's analyze the coefficients of x and y in both equations. In the first equation, the coefficients are 2 and 3, respectively. In the second equation, the coefficients are 1 and 7, respectively.

Solving for x in the First Equation If we solve for x in the first equation, we get:

Isolate 2x 2 x = 18 − 3 y

Solve for x x = 9 − 2 3 ​ y

Solving for x in the Second Equation If we solve for x in the second equation, we get:

Solve for x x = 31 − 7 y

Solving for y in the First Equation If we solve for y in the first equation, we get:

Isolate 3y 3 y = 18 − 2 x

Solve for y y = 6 − 3 2 ​ x

Solving for y in the Second Equation If we solve for y in the second equation, we get:

Isolate 7y 7 y = 31 − x

Solve for y y = 7 31 − x ​

Compare Expressions Comparing the expressions for x and y , we see that solving for x in the second equation is the easiest because it has a coefficient of 1. This avoids fractions in the initial isolation step.

Conclusion Serina is mistaken because solving for x in the second equation is more efficient due to the coefficient of 1. This avoids introducing fractions early in the solving process, making the substitution or elimination steps simpler.


Examples
When solving a system of equations, like determining the intersection point of two lines, efficiency is key. Choosing the right variable to isolate first can significantly simplify the process. For example, if you're managing resources and need to find the optimal allocation point where two different allocation strategies meet, an efficient solution to the system of equations representing these strategies saves time and reduces errors. This approach is also useful in fields like economics, where you might need to find equilibrium points in supply and demand models.

Answered by GinnyAnswer | 2025-07-04