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Sagot :
To solve the system of equations given, we will use a step-by-step approach. The system of equations is:
1. [tex]\(12x + 8y = 4\)[/tex]
2. [tex]\(18x + 10y = 7\)[/tex]
We need to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations.
### Step-by-Step Solution:
1. Define the equations:
[tex]\[ 12x + 8y = 4 \quad \text{(Equation 1)} \][/tex]
[tex]\[ 18x + 10y = 7 \quad \text{(Equation 2)} \][/tex]
2. Simplify if possible:
- Equation 1: [tex]\( 12x + 8y = 4 \)[/tex]
- Equation 2 cannot be simplified in a straightforward manner without altering the given coefficients significantly.
3. Solving using a method of elimination or substitution:
- Let's use the method of elimination:
- Multiply Equation 1 by [tex]\(18\)[/tex] and Equation 2 by [tex]\(12\)[/tex] to make the coefficients of [tex]\(x\)[/tex] the same:
[tex]\[ 18(12x + 8y) = 18(4) \implies 216x + 144y = 72 \quad \text{(Transformed Equation 1)} \][/tex]
[tex]\[ 12(18x + 10y) = 12(7) \implies 216x + 120y = 84 \quad \text{(Transformed Equation 2)} \][/tex]
4. Subtract the transformed equations to eliminate [tex]\(x\)[/tex]:
[tex]\[ (216x + 144y) - (216x + 120y) = 72 - 84 \][/tex]
[tex]\[ 216x + 144y - 216x - 120y = -12 \][/tex]
[tex]\[ 24y = -12 \][/tex]
[tex]\[ y = -\frac{12}{24} = -\frac{1}{2} \][/tex]
5. Substitute the value of [tex]\(y\)[/tex] back into one of the original equations to find [tex]\(x\)[/tex]:
- Substitute [tex]\( y = -\frac{1}{2} \)[/tex] into Equation 1:
[tex]\[ 12x + 8\left(-\frac{1}{2}\right) = 4 \][/tex]
[tex]\[ 12x - 4 = 4 \][/tex]
[tex]\[ 12x = 8 \][/tex]
[tex]\[ x = \frac{8}{12} = \frac{2}{3} \][/tex]
### Conclusion:
The solution to the system of equations is:
[tex]\[ x = \frac{2}{3}, \quad y = -\frac{1}{2} \][/tex]
Therefore, the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both given equations are:
[tex]\[ \left(\frac{2}{3}, -\frac{1}{2}\right) \][/tex]
1. [tex]\(12x + 8y = 4\)[/tex]
2. [tex]\(18x + 10y = 7\)[/tex]
We need to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations.
### Step-by-Step Solution:
1. Define the equations:
[tex]\[ 12x + 8y = 4 \quad \text{(Equation 1)} \][/tex]
[tex]\[ 18x + 10y = 7 \quad \text{(Equation 2)} \][/tex]
2. Simplify if possible:
- Equation 1: [tex]\( 12x + 8y = 4 \)[/tex]
- Equation 2 cannot be simplified in a straightforward manner without altering the given coefficients significantly.
3. Solving using a method of elimination or substitution:
- Let's use the method of elimination:
- Multiply Equation 1 by [tex]\(18\)[/tex] and Equation 2 by [tex]\(12\)[/tex] to make the coefficients of [tex]\(x\)[/tex] the same:
[tex]\[ 18(12x + 8y) = 18(4) \implies 216x + 144y = 72 \quad \text{(Transformed Equation 1)} \][/tex]
[tex]\[ 12(18x + 10y) = 12(7) \implies 216x + 120y = 84 \quad \text{(Transformed Equation 2)} \][/tex]
4. Subtract the transformed equations to eliminate [tex]\(x\)[/tex]:
[tex]\[ (216x + 144y) - (216x + 120y) = 72 - 84 \][/tex]
[tex]\[ 216x + 144y - 216x - 120y = -12 \][/tex]
[tex]\[ 24y = -12 \][/tex]
[tex]\[ y = -\frac{12}{24} = -\frac{1}{2} \][/tex]
5. Substitute the value of [tex]\(y\)[/tex] back into one of the original equations to find [tex]\(x\)[/tex]:
- Substitute [tex]\( y = -\frac{1}{2} \)[/tex] into Equation 1:
[tex]\[ 12x + 8\left(-\frac{1}{2}\right) = 4 \][/tex]
[tex]\[ 12x - 4 = 4 \][/tex]
[tex]\[ 12x = 8 \][/tex]
[tex]\[ x = \frac{8}{12} = \frac{2}{3} \][/tex]
### Conclusion:
The solution to the system of equations is:
[tex]\[ x = \frac{2}{3}, \quad y = -\frac{1}{2} \][/tex]
Therefore, the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both given equations are:
[tex]\[ \left(\frac{2}{3}, -\frac{1}{2}\right) \][/tex]
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