Answer :
Let's solve the given system of linear equations step by step:
[tex]\[ \begin{cases} x - 3y = 15 \quad &(1) \\ y = 5x + 19 \quad &(2) \end{cases} \][/tex]
### Step 1: Substitute Equation (2) into Equation (1)
We have:
[tex]\[ y = 5x + 19 \][/tex]
Substitute [tex]\( y \)[/tex] from Equation (2) into Equation (1):
[tex]\[ x - 3(5x + 19) = 15 \][/tex]
### Step 2: Expand and simplify
First, expand the terms:
[tex]\[ x - 15x - 57 = 15 \][/tex]
Combine like terms:
[tex]\[ -14x - 57 = 15 \][/tex]
### Step 3: Solve for [tex]\( x \)[/tex]
Add 57 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[ -14x = 15 + 57 \][/tex]
[tex]\[ -14x = 72 \][/tex]
Divide both sides by [tex]\(-14\)[/tex]:
[tex]\[ x = \frac{72}{-14} \][/tex]
Simplify the fraction:
[tex]\[ x = -\frac{36}{7} \][/tex]
### Step 4: Substitute [tex]\( x \)[/tex] back into Equation (2) to find [tex]\( y \)[/tex]
Now that we have [tex]\( x = -\frac{36}{7} \)[/tex], we substitute it back into Equation (2):
[tex]\[ y = 5x + 19 \][/tex]
Substitute [tex]\( x \)[/tex] with [tex]\(-\frac{36}{7}\)[/tex]:
[tex]\[ y = 5\left(-\frac{36}{7}\right) + 19 \][/tex]
### Step 5: Simplify the equation for [tex]\( y \)[/tex]
Calculate the multiplication:
[tex]\[ y = -\frac{180}{7} + 19 \][/tex]
Convert 19 to a fraction with the same denominator:
[tex]\[ 19 = \frac{133}{7} \][/tex]
Add the fractions:
[tex]\[ y = -\frac{180}{7} + \frac{133}{7} \][/tex]
Combine the numerators:
[tex]\[ y = \frac{-180 + 133}{7} \][/tex]
[tex]\[ y = -\frac{47}{7} \][/tex]
### Final Solution
Thus, the solution to the system of equations is:
[tex]\[ x = -\frac{36}{7}, \quad y = -\frac{47}{7} \][/tex]
[tex]\[ \begin{cases} x - 3y = 15 \quad &(1) \\ y = 5x + 19 \quad &(2) \end{cases} \][/tex]
### Step 1: Substitute Equation (2) into Equation (1)
We have:
[tex]\[ y = 5x + 19 \][/tex]
Substitute [tex]\( y \)[/tex] from Equation (2) into Equation (1):
[tex]\[ x - 3(5x + 19) = 15 \][/tex]
### Step 2: Expand and simplify
First, expand the terms:
[tex]\[ x - 15x - 57 = 15 \][/tex]
Combine like terms:
[tex]\[ -14x - 57 = 15 \][/tex]
### Step 3: Solve for [tex]\( x \)[/tex]
Add 57 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[ -14x = 15 + 57 \][/tex]
[tex]\[ -14x = 72 \][/tex]
Divide both sides by [tex]\(-14\)[/tex]:
[tex]\[ x = \frac{72}{-14} \][/tex]
Simplify the fraction:
[tex]\[ x = -\frac{36}{7} \][/tex]
### Step 4: Substitute [tex]\( x \)[/tex] back into Equation (2) to find [tex]\( y \)[/tex]
Now that we have [tex]\( x = -\frac{36}{7} \)[/tex], we substitute it back into Equation (2):
[tex]\[ y = 5x + 19 \][/tex]
Substitute [tex]\( x \)[/tex] with [tex]\(-\frac{36}{7}\)[/tex]:
[tex]\[ y = 5\left(-\frac{36}{7}\right) + 19 \][/tex]
### Step 5: Simplify the equation for [tex]\( y \)[/tex]
Calculate the multiplication:
[tex]\[ y = -\frac{180}{7} + 19 \][/tex]
Convert 19 to a fraction with the same denominator:
[tex]\[ 19 = \frac{133}{7} \][/tex]
Add the fractions:
[tex]\[ y = -\frac{180}{7} + \frac{133}{7} \][/tex]
Combine the numerators:
[tex]\[ y = \frac{-180 + 133}{7} \][/tex]
[tex]\[ y = -\frac{47}{7} \][/tex]
### Final Solution
Thus, the solution to the system of equations is:
[tex]\[ x = -\frac{36}{7}, \quad y = -\frac{47}{7} \][/tex]
