Solved on Mar 13, 2024

Solve the system of linear equations $x - 5.1y = -20.3$ and $-x + 5.8y = 22.4$ . The solution is $(\square, \square)$ .

STEP 1

Assumptions
1. We have a system of two linear equations with two variables, $x$ and $y$ .
2. We need to find the values of $x$ and $y$ that satisfy both equations simultaneously.

STEP 2

Write down the system of equations.
\begin{align} x - 5.1y &= -20.3 \quad \text{(Equation 1)} \\ -x + 5.8y &= 22.4 \quad \text{(Equation 2)} \end{align}

STEP 3

To eliminate one of the variables, we can add Equation 1 and Equation 2 together. This will eliminate $x$ because $x$ has a coefficient of 1 in Equation 1 and -1 in Equation 2.

STEP 4

Add Equation 1 and Equation 2.
\begin{align} (x - 5.1y) + (-x + 5.8y) &= -20.3 + 22.4 \end{align}

STEP 5

Simplify the left side of the equation by combining like terms.
\begin{align} x - x - 5.1y + 5.8y &= -20.3 + 22.4 \end{align}

STEP 6

Simplify the right side of the equation by adding the numbers.
\begin{align} 0.7y &= 2.1 \end{align}

STEP 7

Solve for $y$ by dividing both sides of the equation by 0.7.
\begin{align} y &= \frac{2.1}{0.7} \end{align}

STEP 8

Calculate the value of $y$ .
\begin{align} y &= 3 \end{align}

STEP 9

Now that we have the value of $y$ , we can substitute it back into either Equation 1 or Equation 2 to find the value of $x$ . Let's substitute $y$ into Equation 1.

STEP 10

Substitute $y = 3$ into Equation 1.
\begin{align} x - 5.1(3) &= -20.3 \end{align}

STEP 11

Multiply 5.1 by 3.
\begin{align} x - 15.3 &= -20.3 \end{align}

STEP 12

Add 15.3 to both sides of the equation to solve for $x$ .
\begin{align} x &= -20.3 + 15.3 \end{align}

STEP 13

Calculate the value of $x$ .
\begin{align} x &= -5 \end{align}

STEP 14

We have found the values of $x$ and $y$ that satisfy both equations. The solution is the ordered pair $(x, y)$ .
The solution is $(-5, 3)$ .

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Solve the system of linear equations $x - 5.1y = -20.3$ and $-x + 5.8y = 22.4$ . The solution is $(\square, \square)$ .

STEP 1

Assumptions
1. We have a system of two linear equations with two variables, $x$ and $y$ .
2. We need to find the values of $x$ and $y$ that satisfy both equations simultaneously.

STEP 2

Write down the system of equations.
\begin{align} x - 5.1y &= -20.3 \quad \text{(Equation 1)} \\ -x + 5.8y &= 22.4 \quad \text{(Equation 2)} \end{align}

STEP 3

To eliminate one of the variables, we can add Equation 1 and Equation 2 together. This will eliminate $x$ because $x$ has a coefficient of 1 in Equation 1 and -1 in Equation 2.

STEP 4

Add Equation 1 and Equation 2.
\begin{align} (x - 5.1y) + (-x + 5.8y) &= -20.3 + 22.4 \end{align}

STEP 5

Simplify the left side of the equation by combining like terms.
\begin{align} x - x - 5.1y + 5.8y &= -20.3 + 22.4 \end{align}

STEP 6

Simplify the right side of the equation by adding the numbers.
\begin{align} 0.7y &= 2.1 \end{align}

STEP 7

Solve for $y$ by dividing both sides of the equation by 0.7.
\begin{align} y &= \frac{2.1}{0.7} \end{align}

STEP 8

Calculate the value of $y$ .
\begin{align} y &= 3 \end{align}

STEP 9

Now that we have the value of $y$ , we can substitute it back into either Equation 1 or Equation 2 to find the value of $x$ . Let's substitute $y$ into Equation 1.

STEP 10

Substitute $y = 3$ into Equation 1.
\begin{align} x - 5.1(3) &= -20.3 \end{align}

STEP 11

Multiply 5.1 by 3.
\begin{align} x - 15.3 &= -20.3 \end{align}

STEP 12

Add 15.3 to both sides of the equation to solve for $x$ .
\begin{align} x &= -20.3 + 15.3 \end{align}

STEP 13

Calculate the value of $x$ .
\begin{align} x &= -5 \end{align}

STEP 14

We have found the values of $x$ and $y$ that satisfy both equations. The solution is the ordered pair $(x, y)$ .
The solution is $(-5, 3)$ .

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Solve the system of linear equations x−5.1y=−20.3x - 5.1y = -20.3x−5.1y=−20.3 and −x+5.8y=22.4-x + 5.8y = 22.4−x+5.8y=22.4. The solution is (□,□)(\square, \square)(□,□).

STEP 1

Assumptions1. We have a system of two linear equations with two variables, xxx and yyy.2. We need to find the values of xxx and yyy that satisfy both equations simultaneously.

STEP 2

Write down the system of equations.\begin{align*} x - 5.1y &= -20.3 \quad \text{(Equation 1)} \\ -x + 5.8y &= 22.4 \quad \text{(Equation 2)} \end{align*}

STEP 3

To eliminate one of the variables, we can add Equation 1 and Equation 2 together. This will eliminate xxx because xxx has a coefficient of 1 in Equation 1 and -1 in Equation 2.

STEP 4

Add Equation 1 and Equation 2.\begin{align*} (x - 5.1y) + (-x + 5.8y) &= -20.3 + 22.4 \end{align*}

STEP 5

Simplify the left side of the equation by combining like terms.\begin{align*} x - x - 5.1y + 5.8y &= -20.3 + 22.4 \end{align*}

STEP 6

Simplify the right side of the equation by adding the numbers.\begin{align*} 0.7y &= 2.1 \end{align*}

STEP 7

Solve for yyy by dividing both sides of the equation by 0.7.\begin{align*} y &= \frac{2.1}{0.7} \end{align*}

STEP 8

Calculate the value of yyy.\begin{align*} y &= 3 \end{align*}

STEP 9

Now that we have the value of yyy, we can substitute it back into either Equation 1 or Equation 2 to find the value of xxx. Let's substitute yyy into Equation 1.

STEP 10

Substitute y=3y = 3y=3 into Equation 1.\begin{align*} x - 5.1(3) &= -20.3 \end{align*}

STEP 11

Multiply 5.1 by 3.\begin{align*} x - 15.3 &= -20.3 \end{align*}

STEP 12

Add 15.3 to both sides of the equation to solve for xxx.\begin{align*} x &= -20.3 + 15.3 \end{align*}

STEP 13

Calculate the value of xxx.\begin{align*} x &= -5 \end{align*}

STEP 14

We have found the values of xxx and yyy that satisfy both equations. The solution is the ordered pair (x,y)(x, y)(x,y).The solution is (−5,3)(-5, 3)(−5,3).

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Solve the system of linear equations $x - 5.1y = -20.3$ and $-x + 5.8y = 22.4$ . The solution is $(\square, \square)$ .

Assumptions
1. We have a system of two linear equations with two variables, $x$ and $y$ .
2. We need to find the values of $x$ and $y$ that satisfy both equations simultaneously.

Write down the system of equations.
\begin{align} x - 5.1y &= -20.3 \quad \text{(Equation 1)} \\ -x + 5.8y &= 22.4 \quad \text{(Equation 2)} \end{align}

To eliminate one of the variables, we can add Equation 1 and Equation 2 together. This will eliminate $x$ because $x$ has a coefficient of 1 in Equation 1 and -1 in Equation 2.

Add Equation 1 and Equation 2.
\begin{align} (x - 5.1y) + (-x + 5.8y) &= -20.3 + 22.4 \end{align}

Simplify the left side of the equation by combining like terms.
\begin{align} x - x - 5.1y + 5.8y &= -20.3 + 22.4 \end{align}

Simplify the right side of the equation by adding the numbers.
\begin{align} 0.7y &= 2.1 \end{align}

Solve for $y$ by dividing both sides of the equation by 0.7.
\begin{align} y &= \frac{2.1}{0.7} \end{align}

Calculate the value of $y$ .
\begin{align} y &= 3 \end{align}

Now that we have the value of $y$ , we can substitute it back into either Equation 1 or Equation 2 to find the value of $x$ . Let's substitute $y$ into Equation 1.

Substitute $y = 3$ into Equation 1.
\begin{align} x - 5.1(3) &= -20.3 \end{align}

Multiply 5.1 by 3.
\begin{align} x - 15.3 &= -20.3 \end{align}

Add 15.3 to both sides of the equation to solve for $x$ .
\begin{align} x &= -20.3 + 15.3 \end{align}

Calculate the value of $x$ .
\begin{align} x &= -5 \end{align}

We have found the values of $x$ and $y$ that satisfy both equations. The solution is the ordered pair $(x, y)$ .
The solution is $(-5, 3)$ .