Which equation is in standard form?
a. [tex]$x+3=-5 y$[/tex]
b. [tex]$5-y=x$[/tex]
c. [tex]$y=3 x+6$[/tex]
d. [tex]$-8 x+3 y=12$[/tex]
Answer (2)
The standard form of a linear equation is A x + B y = C .
Check each option to see if it is in the form A x + B y = C .
Option d, − 8 x + 3 y = 12 , is already in standard form.
The equation in standard form is − 8 x + 3 y = 12 .
Explanation
Understanding Standard Form We need to identify which of the given equations is in standard form. The standard form of a linear equation is expressed as A x + B y = C , where A , B , and C are constants, and A and B are not both zero. We will examine each option to see if it fits this form.
Analyzing Each Option Let's analyze each option:
a. x + 3 = − 5 y . We can rearrange this to x + 5 y = − 3 . This fits the standard form A x + B y = C , where A = 1 , B = 5 , and C = − 3 .
b. 5 − y = x . Rearranging, we get − x − y = − 5 , which can be written as x + y = 5 . This also fits the standard form, where A = 1 , B = 1 , and C = 5 .
c. y = 3 x + 6 . Rearranging, we get − 3 x + y = 6 . This fits the standard form, where A = − 3 , B = 1 , and C = 6 .
d. − 8 x + 3 y = 12 . This equation is already in the standard form A x + B y = C , where A = − 8 , B = 3 , and C = 12 .
Identifying the Correct Equation The question asks which equation is in standard form without needing rearrangement. Option d, − 8 x + 3 y = 12 , is already in the standard form A x + B y = C .
Conclusion Therefore, the equation that is already in standard form is − 8 x + 3 y = 12 .
Examples
Understanding the standard form of linear equations is crucial in various real-life applications. For instance, when dealing with budget constraints, you might have an equation like 2 x + 5 y = 50 , where x represents the number of items of one type and y represents the number of items of another type you can buy with a total budget of $50. The coefficients 2 and 5 represent the cost of each item, respectively. Recognizing and manipulating equations in standard form helps in solving such problems efficiently.
The equation in standard form is − 8 x + 3 y = 12 . It is already presented in the required form without the need for rearrangement. Options a, b, and c can be rearranged to fit standard form but are not in that format as given.
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