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Sagot :
To solve the given system of equations:
[tex]\[ 5x - 2y = -11 \][/tex]
[tex]\[ -2x + 5y = 17 \][/tex]
Step 1: To create [tex]\(x\)[/tex]-coefficients that are additive inverses, Equation 1 can be multiplied by [tex]\(-2\)[/tex]. Let's proceed step-by-step to understand why and how this multiplication helps in solving the system.
Step 1: Adjust the coefficients of [tex]\(x\)[/tex].
Take Equation 1:
[tex]\[ 5x - 2y = -11 \][/tex]
We want to create [tex]\(x\)[/tex]-coefficients in such a way that they become additive inverses in the two equations. The coefficient of [tex]\(x\)[/tex] in Equation 2 is [tex]\(-2\)[/tex]. If we multiply Equation 1 by [tex]\(-2\)[/tex], we get:
[tex]\[ -2 \cdot (5x - 2y) = -2 \cdot (-11) \][/tex]
This simplifies to:
[tex]\[ -10x + 4y = 22 \][/tex]
So, Equation 1, when multiplied by [tex]\(-2\)[/tex], becomes:
[tex]\[ -10x + 4y = 22 \][/tex]
Step 2: Write down the transformed system of equations.
The transformed system now looks like this:
[tex]\[ -10x + 4y = 22 \][/tex]
[tex]\[ -2x + 5y = 17 \][/tex]
Notice that the coefficients of [tex]\(x\)[/tex] in both equations are not yet additive inverses. To achieve that, we need to similarly adjust the coefficient of [tex]\(x\)[/tex] in Equation 2.
Take Equation 2:
[tex]\[ -2x + 5y = 17 \][/tex]
We need to make the coefficient of [tex]\(x\)[/tex] in Equation 2 additive inverse to the new coefficient of [tex]\(x\)[/tex] coming from Equation 1 multiplied by [tex]\(-2\)[/tex]. Since the Equation 1 was adjusted to [tex]\(-10x\)[/tex], Equation 2 should be adjusted to [tex]\(10x\)[/tex].
Hence, multiply Equation 2 by 5:
[tex]\[ 5 \cdot (-2x + 5y) = 5 \cdot 17 \][/tex]
This gives:
[tex]\[ -10x + 25y = 85 \][/tex]
Step 3: Simplify and combine equations to solve for [tex]\(y\)[/tex].
Now consider the new system of equations:
[tex]\[ -10x + 4y = 22 \][/tex] [tex]\(\quad\)[/tex] (Equation 1 after modification)
[tex]\[ -10x + 25y = 85 \][/tex] [tex]\(\quad\)[/tex] (Equation 2 after modification)
Step 1 has now introduced a common term [tex]\(-10x\)[/tex] in both equations, enabling elimination.
To eliminate [tex]\(x\)[/tex], subtract the first modified equation from the second:
[tex]\[ (-10x + 25y) - (-10x + 4y) = 85 - 22 \][/tex]
This simplifies to:
[tex]\[21y = 63\][/tex]
Solving for [tex]\(y\)[/tex]:
[tex]\[ y = \frac{63}{21} \][/tex]
[tex]\[ y = 3 \][/tex]
Step 4: Substitute [tex]\(y = 3\)[/tex] back into one of the original equations to solve for [tex]\(x\)[/tex].
Substitute [tex]\(y\)[/tex] into Equation 1:
[tex]\[ 5x - 2(3) = -11 \][/tex]
[tex]\[ 5x - 6 = -11 \][/tex]
[tex]\[ 5x = -5 \][/tex]
[tex]\[ x = -1 \][/tex]
So, the solution to the system of equations is:
[tex]\[ x = -1 \][/tex]
[tex]\[ y = 3 \][/tex]
[tex]\[ \boxed{(-1, 3)} \][/tex]
[tex]\[ 5x - 2y = -11 \][/tex]
[tex]\[ -2x + 5y = 17 \][/tex]
Step 1: To create [tex]\(x\)[/tex]-coefficients that are additive inverses, Equation 1 can be multiplied by [tex]\(-2\)[/tex]. Let's proceed step-by-step to understand why and how this multiplication helps in solving the system.
Step 1: Adjust the coefficients of [tex]\(x\)[/tex].
Take Equation 1:
[tex]\[ 5x - 2y = -11 \][/tex]
We want to create [tex]\(x\)[/tex]-coefficients in such a way that they become additive inverses in the two equations. The coefficient of [tex]\(x\)[/tex] in Equation 2 is [tex]\(-2\)[/tex]. If we multiply Equation 1 by [tex]\(-2\)[/tex], we get:
[tex]\[ -2 \cdot (5x - 2y) = -2 \cdot (-11) \][/tex]
This simplifies to:
[tex]\[ -10x + 4y = 22 \][/tex]
So, Equation 1, when multiplied by [tex]\(-2\)[/tex], becomes:
[tex]\[ -10x + 4y = 22 \][/tex]
Step 2: Write down the transformed system of equations.
The transformed system now looks like this:
[tex]\[ -10x + 4y = 22 \][/tex]
[tex]\[ -2x + 5y = 17 \][/tex]
Notice that the coefficients of [tex]\(x\)[/tex] in both equations are not yet additive inverses. To achieve that, we need to similarly adjust the coefficient of [tex]\(x\)[/tex] in Equation 2.
Take Equation 2:
[tex]\[ -2x + 5y = 17 \][/tex]
We need to make the coefficient of [tex]\(x\)[/tex] in Equation 2 additive inverse to the new coefficient of [tex]\(x\)[/tex] coming from Equation 1 multiplied by [tex]\(-2\)[/tex]. Since the Equation 1 was adjusted to [tex]\(-10x\)[/tex], Equation 2 should be adjusted to [tex]\(10x\)[/tex].
Hence, multiply Equation 2 by 5:
[tex]\[ 5 \cdot (-2x + 5y) = 5 \cdot 17 \][/tex]
This gives:
[tex]\[ -10x + 25y = 85 \][/tex]
Step 3: Simplify and combine equations to solve for [tex]\(y\)[/tex].
Now consider the new system of equations:
[tex]\[ -10x + 4y = 22 \][/tex] [tex]\(\quad\)[/tex] (Equation 1 after modification)
[tex]\[ -10x + 25y = 85 \][/tex] [tex]\(\quad\)[/tex] (Equation 2 after modification)
Step 1 has now introduced a common term [tex]\(-10x\)[/tex] in both equations, enabling elimination.
To eliminate [tex]\(x\)[/tex], subtract the first modified equation from the second:
[tex]\[ (-10x + 25y) - (-10x + 4y) = 85 - 22 \][/tex]
This simplifies to:
[tex]\[21y = 63\][/tex]
Solving for [tex]\(y\)[/tex]:
[tex]\[ y = \frac{63}{21} \][/tex]
[tex]\[ y = 3 \][/tex]
Step 4: Substitute [tex]\(y = 3\)[/tex] back into one of the original equations to solve for [tex]\(x\)[/tex].
Substitute [tex]\(y\)[/tex] into Equation 1:
[tex]\[ 5x - 2(3) = -11 \][/tex]
[tex]\[ 5x - 6 = -11 \][/tex]
[tex]\[ 5x = -5 \][/tex]
[tex]\[ x = -1 \][/tex]
So, the solution to the system of equations is:
[tex]\[ x = -1 \][/tex]
[tex]\[ y = 3 \][/tex]
[tex]\[ \boxed{(-1, 3)} \][/tex]
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