8th Grade Math Flashcards: Solve Systems Of Linear Equations

What this deck covers

This deck focuses on Solve Systems Of Linear Equations, giving you a quick way to review the definitions, rules, and examples that matter most for 8th Grade Math.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What condition on slopes and intercepts gives no solution for a linear system?

Answer: Same slope, different $y$-intercepts (parallel distinct lines). Parallel lines never meet, so no point satisfies both.

Flashcard 2: What is the elimination method for solving a system used for?

Answer: Adding or subtracting equations to eliminate one variable. Combine equations to cancel out one variable first.

Flashcard 3: What does the solution of a system of two linear equations represent on a graph?

Answer: The intersection point $(x,y)$ that satisfies both equations. Where two lines cross gives values that make both equations true.

Flashcard 4: What condition on the equations gives infinitely many solutions for a linear system?

Answer: Equivalent equations (the same line), such as multiples of each other. Same line means every point on it satisfies both equations.

Flashcard 5: What is a quick inspection clue that a system has infinitely many solutions?

Answer: One equation simplifies to the other, like $2x+2y=4$ and $x+y=2$. Equivalent equations represent the same line.

Flashcard 6: What is the solution type for y=-rac{1}{2}x+4 and $2y=-x+8$?

Answer: Infinitely many solutions (equivalent equations). Multiply first by $2$: $2y=-x+8$, matching the second.

Flashcard 7: What is the solution type for $y=2x+3$ and $y=2x-1$?

Answer: No solution (parallel lines). Same slope $m=2$ but different $y$-intercepts.

Flashcard 8: What is a quick inspection clue that a system has no solution?

Answer: Same left side but different constants, like $x+y=2$ and $x+y=5$. Can't have same sum equal different values.

Flashcard 9: What is the first step to estimate a system solution by graphing?

Answer: Graph both lines and approximate their intersection point $(x,y)$. Visual method to find where lines meet.

Flashcard 10: What is the solution to the system by inspection: $x+y=7$ and $x+y=7$?

Answer: Infinitely many solutions (the same line). Identical equations describe the same line.

Flashcard 11: What is the solution type by inspection: $2x-3y=6$ and $4x-6y=12$?

Answer: Infinitely many solutions. Second equation is $2×$ the first, so same line.

Flashcard 12: What is the solution type by inspection: $x-2y=1$ and $2x-4y=5$?

Answer: No solution. Second has $2×$ left side but not $2×$ right side.

Flashcard 13: What is the solution to the system: $y=-x+2$ and $y=x-4$?

Answer: $(3,-1)$. Set equal: $-x+2=x-4$, so $2x=6$, thus $x=3$ and $y=-1$.

Flashcard 14: What is the substitution method for solving a system used for?

Answer: Replacing one variable using an equation to solve for the other variable. Substitute to reduce the system to one equation with one variable.

Flashcard 15: What is the solution to the system: $x+y=5$ and $x-y=1$?

Answer: $(3,2)$. Add equations: $2x=6$, so $x=3$; subtract: $2y=4$, so $y=2$.

Flashcard 16: What is the solution to the system: $y=2x+1$ and $y=x+4$?

Answer: $(3,7)$. Set equal: $2x+1=x+4$, so $x=3$ and $y=7$.

Flashcard 17: What are the three possible numbers of solutions for two linear equations in a system?

Answer: $1$ solution, no solution, or infinitely many solutions. Lines can intersect once, never (parallel), or be the same line.

Flashcard 18: What condition on slopes and intercepts gives exactly one solution for two lines?

Answer: Different slopes (not parallel), so the lines intersect once. Non-parallel lines must cross at exactly one point.

Flashcard 19: What is the solution to the system: $3x+2y=12$ and $3x-2y=4$?

Answer: $(\frac{8}{3},2)$. Add equations: $6x=16$, so $x=\frac{8}{3}$; then $y=2$.

Flashcard 20: What is the solution to the system: $2x+y=9$ and $x-y=1$?

Answer: $(\frac{10}{3}, \frac{7}{3})$. Add equations: $3x=10$, so $x=\frac{10}{3}$; then $y=\frac{7}{3}$.

Flashcard 21: Identify the solution by inspection: $y=-x+4$ and $2y=-2x+8$.

Answer: Infinitely many solutions. Second equation is first multiplied by 2, so they're the same line.

Flashcard 22: Solve by elimination: $2x+3y=12$ and $4x+6y=24$.

Answer: Infinitely many solutions. Second equation is first multiplied by 2, same line.

Flashcard 23: Solve by elimination: $2x+3y=12$ and $4x+6y=30$.

Answer: No solution. Second has twice the coefficients but different constant, parallel.

Flashcard 24: What is the approximate solution if two graphed lines intersect at $(2.1,3.9)$?

Answer: $(2.1,3.9)$. The intersection point is the graphical solution.

Flashcard 25: After solving algebraically, what should you do to check the solution $(x,y)$?

Answer: Substitute $(x,y)$ into both equations and verify both are true. Ensures your algebraic solution is correct.

Flashcard 26: Solve by elimination: $x+y=7$ and $x-y=1$.

Answer: $(4,3)$. Add equations: $2x=8$, so $x=4$; subtract: $2y=6$, so $y=3$.

Flashcard 27: Solve by substitution: $y=x+1$ and $2x+y=10$.

Answer: $(3,4)$. Substitute $y=x+1$ into $2x+y=10$: $2x+(x+1)=10$, so $3x=9$.

Flashcard 28: Solve by substitution: $y=2x$ and $x+y=9$.

Answer: $(3,6)$. Substitute $y=2x$ into $x+y=9$: $x+2x=9$, so $3x=9$, $x=3$.

Flashcard 29: Solve by inspection: $x+y=5$ and $x+y=5$.

Answer: Infinitely many solutions. Identical equations represent the same line.

Flashcard 30: Identify the solution by inspection: $y=2x+1$ and $y=2x-3$.

Answer: No solution. Same slope (2) but different $y$-intercepts means parallel lines.