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Sagot :
Certainly! Let’s solve this system of linear equations step by step to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ 4x - 19y + 13 = 0 \quad \text{(Equation 1)} \][/tex]
[tex]\[ 13x - 23y = -19 \quad \text{(Equation 2)} \][/tex]
### Step 1: Rearrange Equation 1
First, we rearrange Equation 1 to isolate the constants on one side:
[tex]\[ 4x - 19y = -13 \][/tex]
### Step 2: Solving the System
We now have a system of equations:
[tex]\[ 4x - 19y = -13 \quad \text{(1)} \][/tex]
[tex]\[ 13x - 23y = -19 \quad \text{(2)} \][/tex]
To eliminate one of the variables, we can use the method of elimination. Let's eliminate [tex]\( x \)[/tex]:
### Step 3: Equalize Coefficients of [tex]\( x \)[/tex]
To eliminate [tex]\( x \)[/tex], we will find a common multiple for the coefficients of [tex]\( x \)[/tex] in both equations. The coefficients are 4 and 13, and the least common multiple (LCM) of [tex]\( 4 \)[/tex] and [tex]\( 13 \)[/tex] is [tex]\( 52 \)[/tex].
Multiply Equation 1 by 13 and Equation 2 by 4:
[tex]\[ 13(4x - 19y) = 13(-13) \implies 52x - 247y = -169 \][/tex]
[tex]\[ 4(13x - 23y) = 4(-19) \implies 52x - 92y = -76 \][/tex]
### Step 4: Subtract the Equations
Now, subtract the second equation from the first:
[tex]\[ (52x - 247y) - (52x - 92y) = -169 - (-76) \][/tex]
This simplifies to:
[tex]\[ 52x - 247y - 52x + 92y = -169 + 76 \][/tex]
[tex]\[ -247y + 92y = -93 \][/tex]
[tex]\[ -155y = -93 \][/tex]
### Step 5: Solve for [tex]\( y \)[/tex]
Now, solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{-93}{-155} = \frac{93}{155} = \frac{3}{5} \][/tex]
### Step 6: Substitute [tex]\( y \)[/tex] back into one of the original equations
Next, substitute [tex]\( y = \frac{3}{5} \)[/tex] back into one of the original equations to solve for [tex]\( x \)[/tex]. Using Equation 1:
[tex]\[ 4x - 19\left(\frac{3}{5}\right) = -13 \][/tex]
[tex]\[ 4x - \frac{57}{5} = -13 \][/tex]
[tex]\[ 4x = -13 + \frac{57}{5} \][/tex]
[tex]\[ 4x = -\frac{65}{5} + \frac{57}{5} \][/tex]
[tex]\[ 4x = -\frac{65 - 57}{5} \][/tex]
[tex]\[ 4x = -\frac{8}{5} \][/tex]
[tex]\[ x = -\frac{8}{5} \times \frac{1}{4} = -\frac{2}{5} \][/tex]
### Final Solutions
The solutions for the variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are:
[tex]\[ x = -\frac{2}{5}, \quad y = \frac{3}{5} \][/tex]
Thus, the solution to the system of equations is:
[tex]\[ \left( x, y \right) = \left( -\frac{2}{5}, \frac{3}{5} \right) \][/tex]
[tex]\[ 4x - 19y + 13 = 0 \quad \text{(Equation 1)} \][/tex]
[tex]\[ 13x - 23y = -19 \quad \text{(Equation 2)} \][/tex]
### Step 1: Rearrange Equation 1
First, we rearrange Equation 1 to isolate the constants on one side:
[tex]\[ 4x - 19y = -13 \][/tex]
### Step 2: Solving the System
We now have a system of equations:
[tex]\[ 4x - 19y = -13 \quad \text{(1)} \][/tex]
[tex]\[ 13x - 23y = -19 \quad \text{(2)} \][/tex]
To eliminate one of the variables, we can use the method of elimination. Let's eliminate [tex]\( x \)[/tex]:
### Step 3: Equalize Coefficients of [tex]\( x \)[/tex]
To eliminate [tex]\( x \)[/tex], we will find a common multiple for the coefficients of [tex]\( x \)[/tex] in both equations. The coefficients are 4 and 13, and the least common multiple (LCM) of [tex]\( 4 \)[/tex] and [tex]\( 13 \)[/tex] is [tex]\( 52 \)[/tex].
Multiply Equation 1 by 13 and Equation 2 by 4:
[tex]\[ 13(4x - 19y) = 13(-13) \implies 52x - 247y = -169 \][/tex]
[tex]\[ 4(13x - 23y) = 4(-19) \implies 52x - 92y = -76 \][/tex]
### Step 4: Subtract the Equations
Now, subtract the second equation from the first:
[tex]\[ (52x - 247y) - (52x - 92y) = -169 - (-76) \][/tex]
This simplifies to:
[tex]\[ 52x - 247y - 52x + 92y = -169 + 76 \][/tex]
[tex]\[ -247y + 92y = -93 \][/tex]
[tex]\[ -155y = -93 \][/tex]
### Step 5: Solve for [tex]\( y \)[/tex]
Now, solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{-93}{-155} = \frac{93}{155} = \frac{3}{5} \][/tex]
### Step 6: Substitute [tex]\( y \)[/tex] back into one of the original equations
Next, substitute [tex]\( y = \frac{3}{5} \)[/tex] back into one of the original equations to solve for [tex]\( x \)[/tex]. Using Equation 1:
[tex]\[ 4x - 19\left(\frac{3}{5}\right) = -13 \][/tex]
[tex]\[ 4x - \frac{57}{5} = -13 \][/tex]
[tex]\[ 4x = -13 + \frac{57}{5} \][/tex]
[tex]\[ 4x = -\frac{65}{5} + \frac{57}{5} \][/tex]
[tex]\[ 4x = -\frac{65 - 57}{5} \][/tex]
[tex]\[ 4x = -\frac{8}{5} \][/tex]
[tex]\[ x = -\frac{8}{5} \times \frac{1}{4} = -\frac{2}{5} \][/tex]
### Final Solutions
The solutions for the variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are:
[tex]\[ x = -\frac{2}{5}, \quad y = \frac{3}{5} \][/tex]
Thus, the solution to the system of equations is:
[tex]\[ \left( x, y \right) = \left( -\frac{2}{5}, \frac{3}{5} \right) \][/tex]
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