Unit 8 Review

Solve$$y=x+4 \qquad y=3x-2$$

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Solve$$2x+y=7 \qquad x-y=2$$

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Check $(1,3)$ in $x+y=4,2x-y=-1$.

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Define break-even.

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Explain why this conclusion is valid: $(2,5)$. Use the foundation problem as evidence: Solve $y=2x+1$ and $y=-x+7$.

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Solve$$y=x^2 \qquad y=4$$

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Solve$$y=x^2 \qquad y=-1$$

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Meaning double root.

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Why graph first?

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Explain why this conclusion is valid: $(1,1)$. Use the foundation problem as evidence: Solve $y=x^2$ and $y=2x-1$.

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Line style $y\le 2x+1$.

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Line style $y>2x+1$.

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Intersection $x+y=8,x+2y=10$.

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Define feasible point.

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Explain why this conclusion is valid: Vertices $(0,0),(0,10),(4,6),(7,0)$. Use the foundation problem as evidence: $x,y\ge 0, x+y\le 10, 2x+y\le 14$.

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One equation in three variables.

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Meaning $0=5$.

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Meaning free variable.

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Solve$$x+y+z=3,x-y+z=1,x+y-z=1$$

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Explain why this conclusion is valid: $(1,2,3)$. Use the foundation problem as evidence: Solve $x+y+z=6,x-y+z=2,x+y-z=0$.

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Dimension with $4 rows,3$ columns.

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Can $2x2$ add $2x3$?

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Dot $(1,3)$ with $(2,-1)$.

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Is AB generally BA?

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Explain why this conclusion is valid: $[[14],[15]]$. Use the foundation problem as evidence: Multiply $[[1,2],[0,3]]$ by $[[4],[5]]$.

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Matrix for $x+y=2,x-y=4$.

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Name row operations.

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Why zero placeholders?

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Meaning divider.

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Explain why this conclusion is valid: $[1,0,3|8]$. Use the foundation problem as evidence: Row for $x+3z=8$ in x,y,z order.

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Define pivot.

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Where zero rows?

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Meaning $[0,0|7]$.

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Can echelon pivots differ from $1$?

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Explain why this conclusion is valid: No solution. Use the foundation problem as evidence: Interpret $[[1,2|3],[0,0|5]]$.

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Is RREF unique?

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Define rank.

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Consistent $2 variables,rank2$.

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Free variables in $5 variables,rank3$.

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Explain why this conclusion is valid: One free variable and infinitely many solutions. Use the foundation problem as evidence: Consistent $3-variable$ system with rank $2$.

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det identity $2x2$.

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det $[[0,1],[1,0]]$.

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What nonzero determinant implies.

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What zero determinant implies.

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Explain why this conclusion is valid: $0$; singular. Use the foundation problem as evidence: det $[[1,2],[2,4]]$.

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Images for $[[0,-1],[1,0]]$.

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Apply to $(3,2)$.

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What transformation?

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Meaning determinant zero.

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Explain why this conclusion is valid: $(7,-3)$. Use the foundation problem as evidence: $A=[[2,1],[0,3]],$ apply to $(4,-1)$.

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