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Sagot :
Sure! Let's break down the solution step-by-step to find the constants that will eliminate one of the variables when the equations are added together.
The given system of equations is:
[tex]\[ \begin{array}{l} 5x + 13y = 232 \\ 12x + 7y = 218 \end{array} \][/tex]
### Step 1: Eliminate [tex]\( y \)[/tex]
To eliminate [tex]\( y \)[/tex], we need to make the coefficients of [tex]\( y \)[/tex] in both equations opposites of each other.
1. Identify the coefficients of [tex]\( y \)[/tex]:
- In the first equation, the coefficient of [tex]\( y \)[/tex] is 13.
- In the second equation, the coefficient of [tex]\( y \)[/tex] is 7.
2. Determine the least common multiple (LCM) of the coefficients:
- The LCM of 13 and 7 is 91.
3. Multiply the equations to make the coefficients of [tex]\( y \)[/tex] opposites:
- To make the coefficient [tex]\( 13y \)[/tex] become [tex]\( 91y \)[/tex], we multiply the first equation by 7.
- To make the coefficient [tex]\( 7y \)[/tex] become [tex]\( -91y \)[/tex], we multiply the second equation by -13.
Thus, the equations should be multiplied as follows:
- The first equation by 7.
- The second equation by -13.
### Step 2: Eliminate [tex]\( x \)[/tex]
To eliminate [tex]\( x \)[/tex], we need to make the coefficients of [tex]\( x \)[/tex] in both equations opposites of each other.
1. Identify the coefficients of [tex]\( x \)[/tex]:
- In the first equation, the coefficient of [tex]\( x \)[/tex] is 5.
- In the second equation, the coefficient of [tex]\( x \)[/tex] is 12.
2. Determine the least common multiple (LCM) of the coefficients:
- The LCM of 5 and 12 is 60.
3. Multiply the equations to make the coefficients of [tex]\( x \)[/tex] opposites:
- To make the coefficient [tex]\( 5x \)[/tex] become [tex]\( 60x \)[/tex], we multiply the first equation by 12.
- To make the coefficient [tex]\( 12x \)[/tex] become [tex]\( -60x \)[/tex], we multiply the second equation by -5.
However, in the given problem, we do not need to use negative signs for this elimination, making it simpler:
- The first equation by 12.
- The second equation by 5.
### Conclusion:
#### To Eliminate [tex]\( y \)[/tex]:
1. The first equation can be multiplied by 7 and the second equation by -13.
2. The first equation can be multiplied by -13 and the second equation by 7 to eliminate [tex]\( y \)[/tex] in a different manner.
#### To Eliminate [tex]\( x \)[/tex]:
1. The first equation can be multiplied by 12 and the second equation by 5.
2. The first equation can be multiplied by -12 and the second equation by 5 to eliminate [tex]\( x \)[/tex] in a different manner.
The given system of equations is:
[tex]\[ \begin{array}{l} 5x + 13y = 232 \\ 12x + 7y = 218 \end{array} \][/tex]
### Step 1: Eliminate [tex]\( y \)[/tex]
To eliminate [tex]\( y \)[/tex], we need to make the coefficients of [tex]\( y \)[/tex] in both equations opposites of each other.
1. Identify the coefficients of [tex]\( y \)[/tex]:
- In the first equation, the coefficient of [tex]\( y \)[/tex] is 13.
- In the second equation, the coefficient of [tex]\( y \)[/tex] is 7.
2. Determine the least common multiple (LCM) of the coefficients:
- The LCM of 13 and 7 is 91.
3. Multiply the equations to make the coefficients of [tex]\( y \)[/tex] opposites:
- To make the coefficient [tex]\( 13y \)[/tex] become [tex]\( 91y \)[/tex], we multiply the first equation by 7.
- To make the coefficient [tex]\( 7y \)[/tex] become [tex]\( -91y \)[/tex], we multiply the second equation by -13.
Thus, the equations should be multiplied as follows:
- The first equation by 7.
- The second equation by -13.
### Step 2: Eliminate [tex]\( x \)[/tex]
To eliminate [tex]\( x \)[/tex], we need to make the coefficients of [tex]\( x \)[/tex] in both equations opposites of each other.
1. Identify the coefficients of [tex]\( x \)[/tex]:
- In the first equation, the coefficient of [tex]\( x \)[/tex] is 5.
- In the second equation, the coefficient of [tex]\( x \)[/tex] is 12.
2. Determine the least common multiple (LCM) of the coefficients:
- The LCM of 5 and 12 is 60.
3. Multiply the equations to make the coefficients of [tex]\( x \)[/tex] opposites:
- To make the coefficient [tex]\( 5x \)[/tex] become [tex]\( 60x \)[/tex], we multiply the first equation by 12.
- To make the coefficient [tex]\( 12x \)[/tex] become [tex]\( -60x \)[/tex], we multiply the second equation by -5.
However, in the given problem, we do not need to use negative signs for this elimination, making it simpler:
- The first equation by 12.
- The second equation by 5.
### Conclusion:
#### To Eliminate [tex]\( y \)[/tex]:
1. The first equation can be multiplied by 7 and the second equation by -13.
2. The first equation can be multiplied by -13 and the second equation by 7 to eliminate [tex]\( y \)[/tex] in a different manner.
#### To Eliminate [tex]\( x \)[/tex]:
1. The first equation can be multiplied by 12 and the second equation by 5.
2. The first equation can be multiplied by -12 and the second equation by 5 to eliminate [tex]\( x \)[/tex] in a different manner.
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