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Solve by Elimination: \( \begin{array}{l}8 x+y=-16 \\ -3 x+y=-5\end{array} \)

Ask by Pena Mcguire. in Vietnam
Jan 21,2025

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Answer

Ratkaisu on \( x = -1 \) ja \( y = -8 \).

Solution

Ratkaistaan yhtälöpari eliminointimenetelmällä: \[ \begin{cases} 8x + y = -16 \\ -3x + y = -5 \end{cases} \] **Vaihe 1:** Vähennetään toinen yhtälö ensimmäisestä yhtälöstä, jotta saadaan eliminoitua muuttuja \( y \). \[ (8x + y) - (-3x + y) = -16 - (-5) \] \[ 8x + y + 3x - y = -16 + 5 \] \[ 11x = -11 \] **Vaihe 2:** Ratkaistaan \( x \): \[ x = \frac{-11}{11} = -1 \] **Vaihe 3:** Sijoitetaan \( x = -1 \) toiseen yhtälöön selvittääksemme \( y \): \[ -3(-1) + y = -5 \] \[ 3 + y = -5 \] \[ y = -5 - 3 \] \[ y = -8 \] **Ratkaisu:** \[ x = -1 \\ y = -8 \]

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To solve the system of equations by elimination, we first want to eliminate one of the variables. Let's eliminate \(y\) by aligning the equations: 1) \(8x + y = -16\) 2) \(-3x + y = -5\) Next, we can subtract the second equation from the first to eliminate \(y\): \[ (8x + y) - (-3x + y) = -16 - (-5) \] This simplifies to: \[ 8x + 3x = -16 + 5 \] \[ 11x = -11 \] Now, solve for \(x\): \[ x = -1 \] Next, substitute \(x = -1\) back into one of the original equations to find \(y\). We'll use the second equation: \[ -3(-1) + y = -5 \] This simplifies to: \[ 3 + y = -5 \] Now, solve for \(y\): \[ y = -5 - 3 \] \[ y = -8 \] Thus, the solution to the system of equations is: \[ x = -1, \quad y = -8 \] So, the solution to the system of equations is \((-1, -8)\).

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