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Solve it! TT \( \sqrt{ \pm} \) Here is the system we created on the last slide.... Solve it! \[ \begin{array}{l} 30 x+4 y=1,040 \\ 12 x+16 y=2,000 \end{array} \] Write your answer as an ordered pair ( \( x, y \) ) Explain the meaning of your solution using a sentence or two. \( \sqrt{ \pm} \) Submit

Ask by Kelley Deleon. in the United States
Feb 26,2025

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The solution is \( (20, 110) \). This means when \( x = 20 \) and \( y = 110 \), both equations are true.

Solución

Solve the system of equations \( 30x+4y=1040;12x+16y=2000 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}30x+4y=1040\\12x+16y=2000\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x=\frac{520-2y}{15}\\12x+16y=2000\end{array}\right.\) - step2: Substitute the value of \(x:\) \(12\times \frac{520-2y}{15}+16y=2000\) - step3: Multiply the terms: \(\frac{4\left(520-2y\right)}{5}+16y=2000\) - step4: Multiply both sides of the equation by LCD: \(\left(\frac{4\left(520-2y\right)}{5}+16y\right)\times 5=2000\times 5\) - step5: Simplify the equation: \(2080+72y=10000\) - step6: Move the constant to the right side: \(72y=10000-2080\) - step7: Subtract the numbers: \(72y=7920\) - step8: Divide both sides: \(\frac{72y}{72}=\frac{7920}{72}\) - step9: Divide the numbers: \(y=110\) - step10: Substitute the value of \(y:\) \(x=\frac{520-2\times 110}{15}\) - step11: Calculate: \(x=20\) - step12: Calculate: \(\left\{ \begin{array}{l}x=20\\y=110\end{array}\right.\) - step13: Check the solution: \(\left\{ \begin{array}{l}x=20\\y=110\end{array}\right.\) - step14: Rewrite: \(\left(x,y\right) = \left(20,110\right)\) The solution to the system of equations is \( (x, y) = (20, 110) \). This means that when \( x = 20 \) and \( y = 110 \), both equations are satisfied. In other words, the point (20, 110) is a solution to the system of equations.

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To solve the system of equations, we can use either the substitution or elimination method. Let's use elimination: 1. **First Equation**: \(30x + 4y = 1,040\) 2. **Second Equation**: \(12x + 16y = 2,000\) To eliminate \(y\), we can multiply the first equation by 4 to align the coefficients of \(y\): \[ 4(30x + 4y) = 4(1,040) \\ 120x + 16y = 4,160 \] Now we have: 1. \(120x + 16y = 4,160\) 2. \(12x + 16y = 2,000\) Now, we subtract the second equation from the first: \[ (120x + 16y) - (12x + 16y) = 4,160 - 2,000 \\ 108x = 2,160 \\ x = \frac{2,160}{108} = 20 \] Now, substitute \(x = 20\) back into one of the original equations, let's use the first one: \[ 30(20) + 4y = 1,040 \\ 600 + 4y = 1,040 \\ 4y = 1,040 - 600 \\ 4y = 440 \\ y = \frac{440}{4} = 110 \] Thus, the solution is the ordered pair \((20, 110)\). Interpreting the solution, this means that at \(x = 20\) and \(y = 110\), both equations balance perfectly, which could represent a scenario such as the allocation of resources or units produced in a real-world application, like a factory's output in a given timeframe.

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