To eliminate the x terms and solve for y in the fewest steps, by which constants should the equations be multiplied by before adding the equations together?

First equation:

Second equation:

The first equation should be multiplied by 3 and the second equation by -5.
The first equation should be multiplied by 3 and the second equation by 5.
The first equation should be multiplied by 7 and the second equation by -6.
The first equation should be multiplied by 7 and the second equation by 6.

To determine the correct constants by which the equations should be multiplied to eliminate the \( x \) terms, we need to consider the coefficients of \( x \) in both equations.

However, since the specific equations aren't provided, I'll describe the general method:

1. Assume the first equation has the form \( ax + by = c \).
2. Assume the second equation has the form \( dx + ey = f \).

To eliminate \( x \) by adding the two equations, we want the coefficients of \( x \) to be equal in magnitude but opposite in sign.

This means we need to find constants \( k_1 \) and \( k_2 \) such that:

\[ k_1 \cdot a = -k_2 \cdot d \]

The options provided suggest specific pairs of constants:

1. The first equation should be multiplied by 3 and the second by \(-5\).
2. The first equation should be multiplied by 3 and the second by 5.
3. The first equation should be multiplied by 7 and the second by \(-6\).
4. The first equation should be multiplied by 7 and the second by 6.

To choose the correct pair, the magnitudes of the coefficients after multiplication should be the same, but with opposite signs.

- Option 1: \( 3a = -5d \) → \( a \) and \( d \) must satisfy \( 3:5 \) ratio with opposite signs.
- Option 2: \( 3a = 5d \) → \( a \) and \( d \) must satisfy \( 3:5 \) ratio with the same sign.
- Option 3: \( 7a = -6d \) → \( a \) and \( d \) must satisfy \( 7:6 \) ratio with opposite signs.
- Option 4: \( 7a = 6d \) → \( a \) and \( d \) must satisfy \( 7:6 \) ratio with the same sign.

For elimination, we want opposite signs. Therefore, the correct options would be:

- Option 1: Multiply the first equation by 3 and the second by \(-5\).
- Option 3: Multiply the first equation by 7 and the second by \(-6\).

Thus, the answer is either Option 1 or Option 3. Without specific values for \( a \) and \( d \), these are the correct methods.

Do you have the specific equations, or would you like further clarification on this process? 

Here are five related questions to expand on this topic:
1. What are the specific steps to solve a system of equations using the elimination method?
2. How can the elimination method be used if the equations have fractions as coefficients?
3. What happens if the equations in a system are already multiples of each other?
4. How do you handle elimination if both coefficients of \( y \) need adjustment?
5. How does the elimination method compare to the substitution method for solving systems of equations?

**Tip:** Always double-check the signs and magnitudes of coefficients when using elimination to ensure the variables cancel correctly.

Learn how to eliminate x terms in a system of equations and solve for y in the fewest steps. Understand which constants to multiply the equations by to achieve elimination. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation