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Caprice adds equations A and B to solve this system of equations. What makes this approach a valid method in general for solving a system of equations?
The method will give the correct solution, x = 3 and y = 1.
8(3) + 3(1) = 27
4(3)–3(1) = 9
Because the y-terms cancel:
(8x + 3y) + (4x–3y) = 8x + 4x + 3y–3y = 12x
As long as addition is used on both sides or subtraction is used on both sides, a true equation will remain true.
Adding a quantity like 8x + 3y to one side of an equation and another quantity like 27 to the other side maintains the equality if the two quantities are equal.
Adding equations A and B is a valid method for solving a system of equations because it preserves the equality of the equations. When you add or subtract the same quantity from both sides of an equation, the relationship remains unchanged, meaning that if the original equations are true, the new equation formed by the addition or subtraction will also be true.
In this specific example, Caprice combines equations that represent two lines in a plane. By adding them, she eliminates the y-terms, simplifying the system to a single-variable equation. This helps in finding the value of x first, leading to the correct solution for both variables. This method works effectively as long as the operations applied are balanced on both sides of the equations.
The approach Caprice uses is valid because when you add or subtract the same quantities from both sides of an equation, you maintain the equality of the equation. This principle is known as the “Equality Property of Addition/Subtraction.” In this case, by combining the equations, Caprice is effectively eliminating one variable (the y-terms), allowing for easier solving of the remaining variable. As a result, the correctness of the solution (x = 3 and y = 1) is preserved throughout the process.
Adding equations A and B is a valid method for solving a system of equations because it maintains the equality. When you add or subtract the same quantity from both sides of an equation, the equation remains true. This means that if both equations describe the same relationship (the same variables), their sum will also describe a valid relationship.
In your example, since the y-terms cancel during the addition, you simplify the problem, allowing you to isolate x. Thus, this method effectively reduces the complexity of the system without losing any information about the solution. If done correctly, it will lead you to the actual values of the variables, ensuring you get the correct solution, x = 3 and y = 1.
The approach Caprice uses is valid because it relies on the fundamental property of equations: if you add or subtract the same quantity from both sides, the equality is preserved. This means that the relationships described by the equations remain unchanged, allowing you to eliminate one variable (in this case, the y-terms) and simplify the system.
When equations A and B are added together, the operation combines their expressions while maintaining the truth of the equations. This makes it easier to solve for the remaining variable (x, in this instance) and ultimately leads to the correct solution.
So in summary, the addition of equations is valid because it keeps the balance of the equations intact, enabling effective elimination of variables. Keep up the good work! If you need more in-depth explanations, feel free to check the extended services page!