Using the equation $y=\frac{2}{3} x-5$, describe how to create a system of linear equations with an infinite number of solutions.
Answer (2)
Multiply the given equation y = 3 2 x − 5 by a non-zero constant k .
Distribute k to obtain k y = 3 2 k x − 5 k .
The system of equations y = 3 2 x − 5 and k y = 3 2 k x − 5 k has infinite solutions.
Example: If k = 2 , the system is y = 3 2 x − 5 and 2 y = 3 4 x − 10 .
Explanation
Understanding Infinite Solutions To create a system of linear equations with an infinite number of solutions from the given equation y = 3 2 x − 5 , we need to form a second equation that is dependent on the first. This means the second equation must represent the same line as the first equation.
Multiplying by a Constant A simple way to achieve this is to multiply the entire equation by a non-zero constant. Let's multiply the given equation by a constant k :
k ( y ) = k ( 3 2 x − 5 )
Distributing the Constant Distribute k on both sides of the equation:
k y = 3 2 k x − 5 k
Forming the System of Equations Now we have a new equation that is a multiple of the original. The system of equations with infinite solutions is:
y = 3 2 x − 5 k y = 3 2 k x − 5 k
For example, if k = 2 , the second equation becomes:
2 y = 3 4 x − 10
Another Example Another example, if k = − 1 , the second equation becomes:
− y = − 3 2 x + 5
Conclusion In summary, to create a system of linear equations with an infinite number of solutions, take the original equation and multiply it by any non-zero constant k . This will result in a second equation that, when paired with the original, forms a dependent system.
Examples
Imagine you're adjusting a recipe. The original recipe calls for certain ratios of ingredients. If you double the entire recipe (multiply all ingredients by 2), you're creating a system with infinite solutions because the ratios remain the same, and the taste will be consistent whether you make a small or large batch. Similarly, in linear equations, multiplying by a constant maintains the same relationship between variables, leading to infinite solutions.
To create a system of linear equations with infinite solutions using the equation y = 3 2 x − 5 , multiply the original equation by any non-zero constant k . This results in a second equation that is dependent on the first, ensuring both equations represent the same line. For example, if k = 2 , the system becomes y = 3 2 x − 5 and 2 y = 3 4 x − 10 .
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