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]
