To solve the system of equations by elimination, the goal is to eliminate one variable by making the coefficients of either $x$ or $y$ the same (or opposites) so that adding or subtracting the equations removes one of the variables.

Here’s the analysis of each option:

1. Multiply the first equation by 3, multiply the second equation by 4, then add the resulting equations together.

• If we do this:
  • Multiply $5x + 4y = 7$ by 3: $15x + 12y = 21$
  • Multiply $2x - 3y = 3$ by 4: $8x - 12y = 12$
  • Adding these eliminates $y$: $15x + 8x = 23x$, and $12y - 12y = 0$.
• This is a valid first step, as $y$ is successfully eliminated.

2. Substitute $3y + 3$ for $x$ in the second equation.

• This is not an elimination step but substitution, which is a completely different method.

3. Substitute $-10y + 7$ for $x$ in the second equation.

• This is also substitution, not elimination.

4. Multiply the first equation by -2, multiply the second equation by -5, then add the resulting equations together.

• If we do this:
  • Multiply $5x + 4y = 7$ by -2: $-10x - 8y = -14$
  • Multiply $2x - 3y = 3$ by -5: $-10x + 15y = -15$
  • Adding these eliminates $x$: $-10x - 10x = 0$, leaving $-8y + 15y = 7y$.
• This is also a valid elimination step.

Correct Answer:

Both Option 1 and Option 4 are valid elimination steps, but Option 1 is typically preferred because it leads directly to the elimination of $y$ (simplifying the system quickly).

Do you have questions or want me to explain the next steps?

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5 Related Questions:

1. How do you decide whether to eliminate $x$ or $y$ first when solving a system of equations?
2. What is substitution, and how does it compare to elimination in solving systems of equations?
3. What happens if the coefficients do not align for easy elimination? How can you modify the equations?
4. Can you solve the system of equations $5x + 4y = 7$ and $2x - 3y = 3$ completely?
5. When might you prefer elimination over substitution or graphing for solving a system of equations?

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Tip:

Always check the resulting equations after elimination to ensure you've eliminated one variable and not introduced errors in your arithmetic!