## Problem: Classify each system and describe its solution For each system of linear equations, compare the two lines. ### Useful rule 1. If the lines have the same slope and the same intercept, they are the **same line** $ \Rightarrow$ **consistent dependent** $\Rightarrow$ **infinitely many solutions**. 2. If the lines have the same slope but different intercepts, they are **parallel** $\Rightarrow$ **inconsistent** $\Rightarrow$ **no solution**. 3. If the slopes are different, the lines **intersect once** $\Rightarrow$ **consistent independent** $\Rightarrow$ **exactly one solution**. --- ## System A Line 1: $y=3x-2$ Line 2: $y=3x+2$ ### Step 1: Compare slopes Both lines are in the form $y=mx+b$. Slope of line 1: $m_1=3$ Slope of line 2: $m_2=3$ So the slopes are equal: $m_1=m_2$. ### Step 2: Compare intercepts Intercept of line 1: $b_1=-2$ Intercept of line 2: $b_2=2$ Since $b_1\ne b_2$, the lines are parallel and distinct. ### Conclusion for System A - The system is **inconsistent**. - The system has **no solution**. --- ## System B Line 1: $y=-\frac{3}{2}x+1$ Line 2: $3x+2y=2$ ### Step 1: Substitute line 1 into line 2 Start with: $$3x+2y=2$$ Substitute $y=-\frac{3}{2}x+1$: $$3x+2\left(-\frac{3}{2}x+1\right)=2$$ ### Step 2: Distribute $$3x+2\left(-\frac{3}{2}x\right)+2\left(1\right)=2$$ $$3x-3x+2=2$$ ### Step 3: Simplify $$3x-3x+2=2$$ $$\cancel{3x}-\cancel{3x}+2=2$$ $$2=2$$ Because the statement is always true, the lines match perfectly (infinitely many solutions). ### Conclusion for System B - The system is **consistent dependent**. - The system has **infinitely many solutions**. - (No single unique solution pair.) --- ## System C Line 1: $y=-x-1$ Line 2: $y=x-1$ ### Step 1: Set the two expressions for $y$ equal Since both equal $y$: $$-x-1=x-1$$ ### Step 2: Solve for $x$ Add $x$ to both sides: $$-x+ x-1=x- x-1$$ $$\cancel{-x}+\cancel{x}-1= x-x-1$$ $$-1=-1$$ This means we made an error in the algebra step if it becomes an identity immediately; instead, subtract $x$ from both sides carefully: Start again: $$-x-1=x-1$$ Subtract $x$ from both sides: $$-x- x-1=x- x-1$$ $$-2x-1=-1$$ Add $1$ to both sides: $$-2x-1+1=-1+1$$ $$-2x=0$$ Divide both sides by $-2$ (show cancellation): $$\cancel{-2}x=\frac{\cancel{0}}{\cancel{-2}}$$ $$x=0$$ ### Step 3: Solve for $y$ Use $y=-x-1$: $$y=-0-1$$ $$y=-1$$ ### Conclusion for System C - The system is **consistent independent**. - The system has **exactly one solution**. - Solution: $\left(0,-1\right)$