Answer :
Sure, let's solve the system of equations step-by-step:
[tex]\[ \begin{cases} 4x + 3y = 23 \\ x - y = 7 \end{cases} \][/tex]
### Step 1: Solve the second equation for [tex]\( x \)[/tex]:
The second equation is:
[tex]\[ x - y = 7 \][/tex]
Add [tex]\( y \)[/tex] to both sides to solve for [tex]\( x \)[/tex]:
[tex]\[ x = y + 7 \][/tex]
### Step 2: Substitute [tex]\( x = y + 7 \)[/tex] into the first equation:
Now we substitute this expression for [tex]\( x \)[/tex] into the first equation:
[tex]\[ 4x + 3y = 23 \][/tex]
[tex]\[ 4(y + 7) + 3y = 23 \][/tex]
### Step 3: Distribute and combine like terms:
Distribute 4 through the [tex]\( y + 7 \)[/tex]:
[tex]\[ 4y + 28 + 3y = 23 \][/tex]
Combine the [tex]\( y \)[/tex] terms:
[tex]\[ 7y + 28 = 23 \][/tex]
### Step 4: Solve for [tex]\( y \)[/tex]:
Subtract 28 from both sides to isolate the [tex]\( y \)[/tex] term:
[tex]\[ 7y = 23 - 28 \][/tex]
[tex]\[ 7y = -5 \][/tex]
Divide both sides by 7:
[tex]\[ y = -\frac{5}{7} \][/tex]
### Step 5: Substitute [tex]\( y \)[/tex] back into [tex]\( x = y + 7 \)[/tex] to find [tex]\( x \)[/tex]:
Now we substitute [tex]\( y = -\frac{5}{7} \)[/tex] back into the expression [tex]\( x = y + 7 \)[/tex]:
[tex]\[ x = -\frac{5}{7} + 7 \][/tex]
Rewrite 7 as a fraction with the same denominator:
[tex]\[ x = -\frac{5}{7} + \frac{49}{7} \][/tex]
Combine the fractions:
[tex]\[ x = \frac{49 - 5}{7} \][/tex]
[tex]\[ x = \frac{44}{7} \][/tex]
So, the solution to the system of equations is:
[tex]\[ x = \frac{44}{7} \][/tex]
[tex]\[ y = -\frac{5}{7} \][/tex]
Therefore, the system of equations:
[tex]\[ \begin{cases} 4x + 3y = 23 \\ x - y = 7 \end{cases} \][/tex]
has the solution:
[tex]\[ \left( x, y \right) = \left( \frac{44}{7}, -\frac{5}{7} \right) \][/tex]
[tex]\[ \begin{cases} 4x + 3y = 23 \\ x - y = 7 \end{cases} \][/tex]
### Step 1: Solve the second equation for [tex]\( x \)[/tex]:
The second equation is:
[tex]\[ x - y = 7 \][/tex]
Add [tex]\( y \)[/tex] to both sides to solve for [tex]\( x \)[/tex]:
[tex]\[ x = y + 7 \][/tex]
### Step 2: Substitute [tex]\( x = y + 7 \)[/tex] into the first equation:
Now we substitute this expression for [tex]\( x \)[/tex] into the first equation:
[tex]\[ 4x + 3y = 23 \][/tex]
[tex]\[ 4(y + 7) + 3y = 23 \][/tex]
### Step 3: Distribute and combine like terms:
Distribute 4 through the [tex]\( y + 7 \)[/tex]:
[tex]\[ 4y + 28 + 3y = 23 \][/tex]
Combine the [tex]\( y \)[/tex] terms:
[tex]\[ 7y + 28 = 23 \][/tex]
### Step 4: Solve for [tex]\( y \)[/tex]:
Subtract 28 from both sides to isolate the [tex]\( y \)[/tex] term:
[tex]\[ 7y = 23 - 28 \][/tex]
[tex]\[ 7y = -5 \][/tex]
Divide both sides by 7:
[tex]\[ y = -\frac{5}{7} \][/tex]
### Step 5: Substitute [tex]\( y \)[/tex] back into [tex]\( x = y + 7 \)[/tex] to find [tex]\( x \)[/tex]:
Now we substitute [tex]\( y = -\frac{5}{7} \)[/tex] back into the expression [tex]\( x = y + 7 \)[/tex]:
[tex]\[ x = -\frac{5}{7} + 7 \][/tex]
Rewrite 7 as a fraction with the same denominator:
[tex]\[ x = -\frac{5}{7} + \frac{49}{7} \][/tex]
Combine the fractions:
[tex]\[ x = \frac{49 - 5}{7} \][/tex]
[tex]\[ x = \frac{44}{7} \][/tex]
So, the solution to the system of equations is:
[tex]\[ x = \frac{44}{7} \][/tex]
[tex]\[ y = -\frac{5}{7} \][/tex]
Therefore, the system of equations:
[tex]\[ \begin{cases} 4x + 3y = 23 \\ x - y = 7 \end{cases} \][/tex]
has the solution:
[tex]\[ \left( x, y \right) = \left( \frac{44}{7}, -\frac{5}{7} \right) \][/tex]
