Solve Systems of Linear Equations - 8th Grade Math
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What condition on slopes and intercepts gives no solution for a linear system?
What condition on slopes and intercepts gives no solution for a linear system?
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Same slope, different $y$-intercepts (parallel distinct lines). Parallel lines never meet, so no point satisfies both.
Same slope, different $y$-intercepts (parallel distinct lines). Parallel lines never meet, so no point satisfies both.
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What is the elimination method for solving a system used for?
What is the elimination method for solving a system used for?
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Adding or subtracting equations to eliminate one variable. Combine equations to cancel out one variable first.
Adding or subtracting equations to eliminate one variable. Combine equations to cancel out one variable first.
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What does the solution of a system of two linear equations represent on a graph?
What does the solution of a system of two linear equations represent on a graph?
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The intersection point $(x,y)$ that satisfies both equations. Where two lines cross gives values that make both equations true.
The intersection point $(x,y)$ that satisfies both equations. Where two lines cross gives values that make both equations true.
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What condition on the equations gives infinitely many solutions for a linear system?
What condition on the equations gives infinitely many solutions for a linear system?
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Equivalent equations (the same line), such as multiples of each other. Same line means every point on it satisfies both equations.
Equivalent equations (the same line), such as multiples of each other. Same line means every point on it satisfies both equations.
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What is a quick inspection clue that a system has infinitely many solutions?
What is a quick inspection clue that a system has infinitely many solutions?
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One equation simplifies to the other, like $2x+2y=4$ and $x+y=2$. Equivalent equations represent the same line.
One equation simplifies to the other, like $2x+2y=4$ and $x+y=2$. Equivalent equations represent the same line.
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What is the solution type for $y=- rac{1}{2}x+4$ and $2y=-x+8$?
What is the solution type for $y=- rac{1}{2}x+4$ and $2y=-x+8$?
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Infinitely many solutions (equivalent equations). Multiply first by $2$: $2y=-x+8$, matching the second.
Infinitely many solutions (equivalent equations). Multiply first by $2$: $2y=-x+8$, matching the second.
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What is the solution type for $y=2x+3$ and $y=2x-1$?
What is the solution type for $y=2x+3$ and $y=2x-1$?
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No solution (parallel lines). Same slope $m=2$ but different $y$-intercepts.
No solution (parallel lines). Same slope $m=2$ but different $y$-intercepts.
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What is a quick inspection clue that a system has no solution?
What is a quick inspection clue that a system has no solution?
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Same left side but different constants, like $x+y=2$ and $x+y=5$. Can't have same sum equal different values.
Same left side but different constants, like $x+y=2$ and $x+y=5$. Can't have same sum equal different values.
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What is the first step to estimate a system solution by graphing?
What is the first step to estimate a system solution by graphing?
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Graph both lines and approximate their intersection point $(x,y)$. Visual method to find where lines meet.
Graph both lines and approximate their intersection point $(x,y)$. Visual method to find where lines meet.
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What is the solution to the system by inspection: $x+y=7$ and $x+y=7$?
What is the solution to the system by inspection: $x+y=7$ and $x+y=7$?
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Infinitely many solutions (the same line). Identical equations describe the same line.
Infinitely many solutions (the same line). Identical equations describe the same line.
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What is the solution type by inspection: $2x-3y=6$ and $4x-6y=12$?
What is the solution type by inspection: $2x-3y=6$ and $4x-6y=12$?
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Infinitely many solutions. Second equation is $2×$ the first, so same line.
Infinitely many solutions. Second equation is $2×$ the first, so same line.
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What is the solution type by inspection: $x-2y=1$ and $2x-4y=5$?
What is the solution type by inspection: $x-2y=1$ and $2x-4y=5$?
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No solution. Second has $2×$ left side but not $2×$ right side.
No solution. Second has $2×$ left side but not $2×$ right side.
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What is the solution to the system: $y=-x+2$ and $y=x-4$?
What is the solution to the system: $y=-x+2$ and $y=x-4$?
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$(3,-1)$. Set equal: $-x+2=x-4$, so $2x=6$, thus $x=3$ and $y=-1$.
$(3,-1)$. Set equal: $-x+2=x-4$, so $2x=6$, thus $x=3$ and $y=-1$.
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What is the substitution method for solving a system used for?
What is the substitution method for solving a system used for?
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Replacing one variable using an equation to solve for the other variable. Substitute to reduce the system to one equation with one variable.
Replacing one variable using an equation to solve for the other variable. Substitute to reduce the system to one equation with one variable.
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What is the solution to the system: $x+y=5$ and $x-y=1$?
What is the solution to the system: $x+y=5$ and $x-y=1$?
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$(3,2)$. Add equations: $2x=6$, so $x=3$; subtract: $2y=4$, so $y=2$.
$(3,2)$. Add equations: $2x=6$, so $x=3$; subtract: $2y=4$, so $y=2$.
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What is the solution to the system: $y=2x+1$ and $y=x+4$?
What is the solution to the system: $y=2x+1$ and $y=x+4$?
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$(3,7)$. Set equal: $2x+1=x+4$, so $x=3$ and $y=7$.
$(3,7)$. Set equal: $2x+1=x+4$, so $x=3$ and $y=7$.
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What are the three possible numbers of solutions for two linear equations in a system?
What are the three possible numbers of solutions for two linear equations in a system?
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$1$ solution, no solution, or infinitely many solutions. Lines can intersect once, never (parallel), or be the same line.
$1$ solution, no solution, or infinitely many solutions. Lines can intersect once, never (parallel), or be the same line.
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What condition on slopes and intercepts gives exactly one solution for two lines?
What condition on slopes and intercepts gives exactly one solution for two lines?
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Different slopes (not parallel), so the lines intersect once. Non-parallel lines must cross at exactly one point.
Different slopes (not parallel), so the lines intersect once. Non-parallel lines must cross at exactly one point.
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What is the solution to the system: $3x+2y=12$ and $3x-2y=4$?
What is the solution to the system: $3x+2y=12$ and $3x-2y=4$?
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$(\frac{8}{3},2)$. Add equations: $6x=16$, so $x=\frac{8}{3}$; then $y=2$.
$(\frac{8}{3},2)$. Add equations: $6x=16$, so $x=\frac{8}{3}$; then $y=2$.
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What is the solution to the system: $2x+y=9$ and $x-y=1$?
What is the solution to the system: $2x+y=9$ and $x-y=1$?
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$(\frac{10}{3}, \frac{7}{3})$. Add equations: $3x=10$, so $x=\frac{10}{3}$; then $y=\frac{7}{3}$.
$(\frac{10}{3}, \frac{7}{3})$. Add equations: $3x=10$, so $x=\frac{10}{3}$; then $y=\frac{7}{3}$.
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Identify the solution by inspection: $y=-x+4$ and $2y=-2x+8$.
Identify the solution by inspection: $y=-x+4$ and $2y=-2x+8$.
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Infinitely many solutions. Second equation is first multiplied by 2, so they're the same line.
Infinitely many solutions. Second equation is first multiplied by 2, so they're the same line.
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Solve by elimination: $2x+3y=12$ and $4x+6y=24$.
Solve by elimination: $2x+3y=12$ and $4x+6y=24$.
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Infinitely many solutions. Second equation is first multiplied by 2, same line.
Infinitely many solutions. Second equation is first multiplied by 2, same line.
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Solve by elimination: $2x+3y=12$ and $4x+6y=30$.
Solve by elimination: $2x+3y=12$ and $4x+6y=30$.
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No solution. Second has twice the coefficients but different constant, parallel.
No solution. Second has twice the coefficients but different constant, parallel.
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What is the approximate solution if two graphed lines intersect at $(2.1,3.9)$?
What is the approximate solution if two graphed lines intersect at $(2.1,3.9)$?
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$(2.1,3.9)$. The intersection point is the graphical solution.
$(2.1,3.9)$. The intersection point is the graphical solution.
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After solving algebraically, what should you do to check the solution $(x,y)$?
After solving algebraically, what should you do to check the solution $(x,y)$?
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Substitute $(x,y)$ into both equations and verify both are true. Ensures your algebraic solution is correct.
Substitute $(x,y)$ into both equations and verify both are true. Ensures your algebraic solution is correct.
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Solve by elimination: $x+y=7$ and $x-y=1$.
Solve by elimination: $x+y=7$ and $x-y=1$.
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$(4,3)$. Add equations: $2x=8$, so $x=4$; subtract: $2y=6$, so $y=3$.
$(4,3)$. Add equations: $2x=8$, so $x=4$; subtract: $2y=6$, so $y=3$.
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Solve by substitution: $y=x+1$ and $2x+y=10$.
Solve by substitution: $y=x+1$ and $2x+y=10$.
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$(3,4)$. Substitute $y=x+1$ into $2x+y=10$: $2x+(x+1)=10$, so $3x=9$.
$(3,4)$. Substitute $y=x+1$ into $2x+y=10$: $2x+(x+1)=10$, so $3x=9$.
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Solve by substitution: $y=2x$ and $x+y=9$.
Solve by substitution: $y=2x$ and $x+y=9$.
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$(3,6)$. Substitute $y=2x$ into $x+y=9$: $x+2x=9$, so $3x=9$, $x=3$.
$(3,6)$. Substitute $y=2x$ into $x+y=9$: $x+2x=9$, so $3x=9$, $x=3$.
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Solve by inspection: $x+y=5$ and $x+y=5$.
Solve by inspection: $x+y=5$ and $x+y=5$.
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Infinitely many solutions. Identical equations represent the same line.
Infinitely many solutions. Identical equations represent the same line.
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Identify the solution by inspection: $y=2x+1$ and $y=2x-3$.
Identify the solution by inspection: $y=2x+1$ and $y=2x-3$.
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No solution. Same slope (2) but different $y$-intercepts means parallel lines.
No solution. Same slope (2) but different $y$-intercepts means parallel lines.
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