Using the substitution method, what is the solution to the following system of equations?

[tex]2 x-4 y=10 ; 2 y=x-5[/tex]

Express x in terms of y from the second equation: x = 2 y + 5 .
Substitute this expression for x into the first equation: 2 ( 2 y + 5 ) − 4 y = 10 .
Simplify the equation: 4 y + 10 − 4 y = 10 , which simplifies to 10 = 10 .
Since the equation is always true, there are infinitely many solutions. A nin f ini t e n u mb ero f so l u t i o n s ​

Explanation

Analyze the problem We are given the system of equations:

2 x − 4 y = 10 (1)
2 y = x − 5 (2)
We will use the substitution method to solve this system.

Express x in terms of y From equation (2), we can express x in terms of y :

x = 2 y + 5 (3)

Substitute into the first equation Substitute equation (3) into equation (1):

2 ( 2 y + 5 ) − 4 y = 10
4 y + 10 − 4 y = 10
10 = 10

Solve for y Since the y terms canceled out and we are left with a true statement ( 10 = 10 ), this means that the two equations are dependent, and there are infinitely many solutions.

Conclusion The system has infinitely many solutions.


Examples
Systems of equations are used in various real-life situations, such as determining the break-even point for a business. For example, if a company has fixed costs and variable costs, and they sell a product at a certain price, they can use a system of equations to find the number of units they need to sell to cover their costs. Similarly, in physics, systems of equations can be used to analyze the forces acting on an object in equilibrium. Understanding how to solve systems of equations is a fundamental skill that has broad applications across many disciplines.

Answered by GinnyAnswer | 2025-07-04

The system of equations has infinitely many solutions since they represent the same line. Any values of x and y that satisfy x = 2y + 5 will work as solutions. This indicates that there are infinitely many pairs of (x, y) that fulfill both equations.
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Answered by Anonymous | 2025-07-11