What will the first step look like for the substitution problem:
$\begin{array}{l}
y=2 x \
x+y=9
\end{array}$
A. $2 y+y=9$
B. $y+2 x+x+y=9$
C. $y=2 x+9$
D. $x+2 x=9$
Answer (2)
Substitute the expression for y from the first equation ( y = 2 x ) into the second equation ( x + y = 9 ).
Replace y in the second equation with 2 x , resulting in x + ( 2 x ) = 9 .
Simplify the equation to x + 2 x = 9 .
The correct first step for solving the system by substitution is x + 2 x = 9 .
Explanation
Understanding the Problem We are given a system of two equations:
y = 2 x x + y = 9
We want to find the correct first step to solve this system using the substitution method. The substitution method involves solving one equation for one variable and substituting that expression into the other equation. In this case, the first equation is already solved for y in terms of x .
Performing the Substitution We can substitute the expression for y from the first equation, y = 2 x , into the second equation, x + y = 9 . This means we replace y in the second equation with 2 x . So, the second equation becomes:
x + ( 2 x ) = 9
Which simplifies to:
x + 2 x = 9
Identifying the Correct Option Now, let's compare this result with the given options:
a. 2 y + y = 9 (Incorrect) b. y + 2 x + x + y = 9 (Incorrect) c. y = 2 x + 9 (Incorrect) d. x + 2 x = 9 (Correct)
The correct first step is x + 2 x = 9 .
Examples
Substitution is a powerful tool in algebra, used to solve systems of equations by expressing one variable in terms of others. Imagine you're baking cookies and the recipe calls for twice as much flour as sugar. If you know you have 3 cups of sugar and flour combined, you can use substitution to figure out exactly how much of each ingredient you need. By letting 'f' represent flour and 's' represent sugar, you have the equations f = 2s and f + s = 3. Substituting 2s for f in the second equation gives 2s + s = 3, which simplifies to 3s = 3, so s = 1 cup of sugar. Then, f = 2 cups of flour. This method helps in many real-world scenarios, from cooking to engineering, where you need to find specific values based on related quantities.
The first step for solving the given system of equations by substitution is to substitute y = 2 x into the second equation, resulting in x + 2 x = 9 . Therefore, the correct option is D, which states x + 2 x = 9 . This process is vital for simplifying the equations and finding the variables' values.
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