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Sagot :
To solve the system of equations
[tex]\[ \left\{\begin{array}{l} 13x - 6y = 2 \\ 3x - 4y = -10 \end{array}\right. \][/tex]
we will use the method of elimination or substitution. Here is a detailed step-by-step solution:
1. Step 1: Multiply the second equation by a factor that will make the coefficients of either [tex]\(x\)[/tex] or [tex]\(y\)[/tex] in both equations the same (or opposites).
We can notice that multiplying the second equation by 2 will make the coefficient of [tex]\(y\)[/tex] in both equations equal (in magnitude):
[tex]\[ 2(3x - 4y) = 2(-10) \][/tex]
which simplifies to:
[tex]\[ 6x - 8y = -20 \][/tex]
2. Step 2: Rewrite the system with the new equation:
[tex]\[ \left\{\begin{array}{l} 13x - 6y = 2 \\ 6x - 8y = -20 \end{array}\right. \][/tex]
3. Step 3: Eliminate one of the variables by combining the equations. To do this, we can multiply the first equation by 4 and the second equation by 3 to align the [tex]\(y\)[/tex] coefficients:
[tex]\[ 4(13x - 6y) = 4(2) \implies 52x - 24y = 8 \][/tex]
[tex]\[ 3(6x - 8y) = 3(-20) \implies 18x - 24y = -60 \][/tex]
4. Step 4: Subtract the second modified equation from the first modified equation to eliminate [tex]\(y\)[/tex]:
[tex]\[ (52x - 24y) - (18x - 24y) = 8 - (-60) \][/tex]
This simplifies to:
[tex]\[ 34x = 68 \][/tex]
5. Step 5: Solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{68}{34} = 2 \][/tex]
6. Step 6: Substitute [tex]\(x = 2\)[/tex] back into one of the original equations to find the value of [tex]\(y\)[/tex]. We use the second original equation for this:
[tex]\[ 3x - 4y = -10 \][/tex]
Substitute [tex]\(x = 2\)[/tex]:
[tex]\[ 3(2) - 4y = -10 \][/tex]
[tex]\[ 6 - 4y = -10 \][/tex]
[tex]\[ -4y = -10 - 6 \][/tex]
[tex]\[ -4y = -16 \][/tex]
7. Step 7: Solve for [tex]\(y\)[/tex]:
[tex]\[ y = \frac{-16}{-4} = 4 \][/tex]
Therefore, the solution to the system of equations is [tex]\(x = 2\)[/tex] and [tex]\(y = 4\)[/tex].
The final answer is:
[tex]\[ \boxed{(2, 4)} \][/tex]
[tex]\[ \left\{\begin{array}{l} 13x - 6y = 2 \\ 3x - 4y = -10 \end{array}\right. \][/tex]
we will use the method of elimination or substitution. Here is a detailed step-by-step solution:
1. Step 1: Multiply the second equation by a factor that will make the coefficients of either [tex]\(x\)[/tex] or [tex]\(y\)[/tex] in both equations the same (or opposites).
We can notice that multiplying the second equation by 2 will make the coefficient of [tex]\(y\)[/tex] in both equations equal (in magnitude):
[tex]\[ 2(3x - 4y) = 2(-10) \][/tex]
which simplifies to:
[tex]\[ 6x - 8y = -20 \][/tex]
2. Step 2: Rewrite the system with the new equation:
[tex]\[ \left\{\begin{array}{l} 13x - 6y = 2 \\ 6x - 8y = -20 \end{array}\right. \][/tex]
3. Step 3: Eliminate one of the variables by combining the equations. To do this, we can multiply the first equation by 4 and the second equation by 3 to align the [tex]\(y\)[/tex] coefficients:
[tex]\[ 4(13x - 6y) = 4(2) \implies 52x - 24y = 8 \][/tex]
[tex]\[ 3(6x - 8y) = 3(-20) \implies 18x - 24y = -60 \][/tex]
4. Step 4: Subtract the second modified equation from the first modified equation to eliminate [tex]\(y\)[/tex]:
[tex]\[ (52x - 24y) - (18x - 24y) = 8 - (-60) \][/tex]
This simplifies to:
[tex]\[ 34x = 68 \][/tex]
5. Step 5: Solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{68}{34} = 2 \][/tex]
6. Step 6: Substitute [tex]\(x = 2\)[/tex] back into one of the original equations to find the value of [tex]\(y\)[/tex]. We use the second original equation for this:
[tex]\[ 3x - 4y = -10 \][/tex]
Substitute [tex]\(x = 2\)[/tex]:
[tex]\[ 3(2) - 4y = -10 \][/tex]
[tex]\[ 6 - 4y = -10 \][/tex]
[tex]\[ -4y = -10 - 6 \][/tex]
[tex]\[ -4y = -16 \][/tex]
7. Step 7: Solve for [tex]\(y\)[/tex]:
[tex]\[ y = \frac{-16}{-4} = 4 \][/tex]
Therefore, the solution to the system of equations is [tex]\(x = 2\)[/tex] and [tex]\(y = 4\)[/tex].
The final answer is:
[tex]\[ \boxed{(2, 4)} \][/tex]
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