Question

this question has two parts. first, answer part a. then, answer part b.
part a
a. write the given equation in standard form.
$-0.08x = 1.24y - 3.12$
b. identify a, b, and c
$a = \boxed{}$
$b = \boxed{}$
$c = \boxed{}$
part b
(calculator interface shown with buttons: +, -, ×, ÷, (), √, ∛, =, ≈, <, >, ≤, ≥, (), n)

Explanation:

Step1: Recall standard form

The standard form of a linear equation is \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(A\geq0\).

Step2: Rearrange the given equation

Given equation: \(-0.08x = 1.24y - 3.12\)
Subtract \(1.24y\) from both sides: \(-0.08x - 1.24y = -3.12\)
Multiply all terms by -100 to eliminate decimals: \(8x + 124y = 312\)
We can also simplify by dividing all terms by 4: \(2x + 31y = 78\) (this is also in standard form, but let's first use the non - simplified integer coefficients from multiplying by - 100 for part B identification, or we can use the simplified one. Let's check the original rearrangement without simplifying the integer coefficients too much first. When we move all terms to the left side from \(-0.08x-1.24y + 3.12=0\) and then multiply by 100 to get rid of decimals: \( - 8x-124y+312 = 0\), then rewrite as \(Ax + By = C\) form: \(-8x-124y=-312\), but we usually want \(A\geq0\), so multiply both sides by - 1: \(8x + 124y = 312\))

For part B, if we take the equation in the form \(Ax+By = C\) as \(8x+124y = 312\) (after multiplying the original equation \(-0.08x-1.24y=-3.12\) by - 100), then:

• \(A = 8\)
• \(B = 124\)
• \(C = 312\)

If we simplify the equation \(8x + 124y = 312\) by dividing each term by 4, we get \(2x+31y = 78\), then:

• \(A = 2\)
• \(B = 31\)
• \(C = 78\)

Part A Answer (standard form):

One possible standard form is \(8x + 124y = 312\) (or the simplified \(2x+31y = 78\))

Part B Answers:

If we use \(8x + 124y = 312\):

• \(A=\boldsymbol{8}\)
• \(B=\boldsymbol{124}\)
• \(C=\boldsymbol{312}\)

If we use the simplified form \(2x + 31y=78\):

• \(A=\boldsymbol{2}\)
• \(B=\boldsymbol{31}\)
• \(C=\boldsymbol{78}\)

(Note: The key is to get the equation into \(Ax + By = C\) form with integer coefficients and \(A\geq0\). The process of multiplying to clear decimals and rearranging gives us the values for \(A\), \(B\), and \(C\))

Answer:

Part A:

The equation in standard form (one possible) is \(8x + 124y = 312\) (or \(2x+31y = 78\))

Part B:

If using \(8x + 124y = 312\): \(A = 8\), \(B = 124\), \(C = 312\)

If using \(2x+31y = 78\): \(A = 2\), \(B = 31\), \(C = 78\)