To determine how many solutions this system of linear equations has, let's analyze the given system and find its solutions step-by-step.
We have the following system of equations:
[tex]\[
\begin{array}{l}
y = \frac{2}{3} x + 2 \\
6x - 4y = -10
\end{array}
\][/tex]
### Step 1: Substitute [tex]\( y \)[/tex] from the first equation into the second equation
Given:
[tex]\[ y = \frac{2}{3} x + 2 \][/tex]
Substitute this expression for [tex]\( y \)[/tex] into the second equation:
[tex]\[ 6x - 4 \left(\frac{2}{3} x + 2 \right) = -10 \][/tex]
### Step 2: Simplify the equation
First, distribute [tex]\( -4 \)[/tex] over the terms in the parentheses:
[tex]\[ 6x - \left( \frac{8}{3} x + 8 \right) = -10 \][/tex]
Next, combine the terms with [tex]\( x \)[/tex]:
[tex]\[ 6x - \frac{8}{3} x - 8 = -10 \][/tex]
Combine like terms (note that [tex]\( 6x \)[/tex] can also be written as [tex]\( \frac{18}{3} x \)[/tex]):
[tex]\[ \frac{18}{3} x - \frac{8}{3} x - 8 = -10 \][/tex]
[tex]\[ \frac{10}{3} x - 8 = -10 \][/tex]
### Step 3: Solve for [tex]\( x \)[/tex]
To isolate [tex]\( x \)[/tex], add [tex]\( 8 \)[/tex] to both sides of the equation:
[tex]\[ \frac{10}{3} x = -2 \][/tex]
Now, multiply both sides by [tex]\( \frac{3}{10} \)[/tex] to solve for [tex]\( x \)[/tex]:
[tex]\[ x = -2 \times \frac{3}{10} \][/tex]
[tex]\[ x = -\frac{6}{10} \][/tex]
[tex]\[ x = -0.6 \][/tex]
### Step 4: Solve for [tex]\( y \)[/tex]
Substitute [tex]\( x = -0.6 \)[/tex] back into the first equation:
[tex]\[ y = \frac{2}{3} (-0.6) + 2 \][/tex]
[tex]\[ y = \frac{2 \times -0.6}{3} + 2 \][/tex]
[tex]\[ y = -0.4 + 2 \][/tex]
[tex]\[ y = 1.6 \][/tex]
### Step 5: Confirm the solution
The calculated values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are:
[tex]\[ x = -0.6 \][/tex]
[tex]\[ y = 1.6 \][/tex]
### Conclusion
Since we have found a unique solution for [tex]\( x \)[/tex] and [tex]\( y \)[/tex], the system of equations has one solution.
The correct answer is:
- one solution: [tex]\((-0.6, 1.6)\)[/tex]
Therefore, the number of solutions to this system of equations is:
one solution: [tex]\((-0.6, 1.6)\)[/tex]
