Select all of the ways the system of equations can be solved: 9x - 2y = 4 3x + 8y = -12 * Multiply the second equation by 3, then add the equations. * Multiply the first equat... Select all of the ways the system of equations can be solved: 9x - 2y = 4 3x + 8y = -12 * Multiply the second equation by 3, then add the equations. * Multiply the first equation by 4, then add the equations. * Multiply the first equation by 4, then subtract the equations. * Multiply the first equation by 3, then add the equations. * Multiply the second equation by 3, then subtract the equations.
Understand the Problem
The question asks to identify valid methods for solving a system of linear equations. Given two equations (9x - 2y = 4 and 3x + 8y = -12), several options involve multiplying one equation by a constant and then adding or subtracting the equations. We need to determine which of these operations would correctly eliminate one variable, allowing us to solve for other variable.
Answer
- Multiply the first equation by 4, then add the equations. - Multiply the second equation by 3, then subtract the equations.
Answer for screen readers
- Multiply the first equation by 4, then add the equations.
- Multiply the second equation by 3, then subtract the equations.
Steps to Solve
- Analyze the first option: Multiply the second equation by 3, then add the equations.
Multiply the second equation $3x + 8y = -12$ by 3: $3(3x + 8y) = 3(-12)$ $9x + 24y = -36$ Add this new equation to the first equation $9x - 2y = 4$: $(9x - 2y) + (9x + 24y) = 4 + (-36)$ $18x + 22y = -32$ This does not eliminate any variable. Therefore, this option is incorrect.
- Analyze the second option: Multiply the first equation by 4, then add the equations.
Multiply the first equation $9x - 2y = 4$ by 4: $4(9x - 2y) = 4(4)$ $36x - 8y = 16$ Add this new equation to the second equation $3x + 8y = -12$: $(36x - 8y) + (3x + 8y) = 16 + (-12)$ $39x = 4$ This eliminates $y$. Therefore, this option is correct.
- Analyze the third option: Multiply the first equation by 4, then subtract the equations.
Multiply the first equation $9x - 2y = 4$ by 4: $4(9x - 2y) = 4(4)$ $36x - 8y = 16$ Subtract the second equation $3x + 8y = -12$ from the modified first equation: $(36x - 8y) - (3x + 8y) = 16 - (-12)$ $33x - 16y = 28$ This does not eliminate any variable. Therefore, this option is incorrect.
- Analyze the fourth option: Multiply the first equation by 3, then add the equations.
Multiply the first equation $9x - 2y = 4$ by 3: $3(9x - 2y) = 3(4)$ $27x - 6y = 12$ Add this new equation to the second equation $3x + 8y = -12$: $(27x - 6y) + (3x + 8y) = 12 + (-12)$ $30x + 2y = 0$ This does not eliminate any variable. Therefore, this option is incorrect.
- Analyze the fifth option: Multiply the second equation by 3, then subtract the equations.
Multiply the second equation $3x + 8y = -12$ by 3: $3(3x + 8y) = 3(-12)$ $9x + 24y = -36$ Subtract this new equation from the first equation $9x - 2y = 4$: $(9x - 2y) - (9x + 24y) = 4 - (-36)$ $-26y = 40$ This eliminates $x$. Therefore, this option is correct.
- Multiply the first equation by 4, then add the equations.
- Multiply the second equation by 3, then subtract the equations.
More Information
The elimination method involves manipulating equations so that when they are added or subtracted, one of the variables is canceled out, making it easier to solve for the remaining variable.
Tips
A common mistake is not distributing the multiplication correctly across all terms in the equation. Another mistake is performing the addition or subtraction incorrectly, especially with negative signs.
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