Explain how to change the following system to an equivalent system with integer coefficients. Write the equivalent system. $$ \begin{aligned} 0.5 x-0.3 y & =0.4 \ 5 x+3 y & =6\end{aligned} $$

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The equivalent system with integer coefficients is: $$ \begin{aligned} 5 x - 3 y & = 4 \ 5 x + 3 y & = 6 \end{aligned} $$ To convert the given system of equations to an equivalent system with integer coefficients, we can eliminate the decimals by multiplying each equation by an appropriate power of 10. The given system is: $$ \begin{aligned} 0.5 x - 0.3 y & = 0.4 \quad \text{(1)} \ 5 x + 3 y & = 6 \quad \text{(2)} \end{aligned} $$ ### Step 1: Eliminate decimals in Equation (1) In Equation (1), the coefficients are $$0.5$$, $$-0.3$$, and the constant term is $$0.4$$. To eliminate the decimals, we can multiply the entire equation by 10: $$ 10(0.5 x) - 10(0.3 y) = 10(0.4) $$ This simplifies to: $$ 5 x - 3 y = 4 \quad \text{(3)} $$ ### Step 2: Check Equation (2) Equation (2) already has integer coefficients: $$ 5 x + 3 y = 6 \quad \text{(2)} $$ ### Step 3: Write the equivalent system Now we can write the equivalent system of equations with integer coefficients: $$ \begin{aligned} 5 x - 3 y & = 4 \quad \text{(from Equation (1))} \ 5 x + 3 y & = 6 \quad \text{(from Equation (2))} \end{aligned} $$ Thus, the equivalent system with integer coefficients is: $$ \begin{aligned} 5 x - 3 y & = 4 \ 5 x + 3 y & = 6 \end{aligned} $$

Explain how to change the following system to an equivalent system with integer coefficients. Write the equivalent system. \( \begin{aligned} 0.5 x-0.3 y & =0.4 \\ 5 x+3 y & =6\end{aligned} \)

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Explain how to change the following system to an equivalent system with integer coefficients. Write the equivalent system. \( \begin{aligned} 0.5 x-0.3 y & =0.4 \\ 5 x+3 y & =6\end{aligned} \)

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