Algebra 2 Flashcards: Solving Systems Of Linear Equations

What this deck covers

This deck focuses on Solving Systems Of Linear Equations, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra 2.

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: Identify the best method for $3x+2y=7$ and $6x-2y=5$ if you want quick elimination.

Answer: Elimination. Coefficients of $y$ are already opposites.

Flashcard 2: What is the solution of the system $x-2y=0$ and $3x+2y=20$?

Answer: $(5,\frac{5}{2})$. Add equations to eliminate $y$: $4x=20$, so $x=5$.

Flashcard 3: What is the solution of the system $x+y=1$ and $x-y=9$?

Answer: $(5,-4)$. Add equations to get $2x=10$, so $x=5$.

Flashcard 4: What does it mean if two lines on a graph do not intersect within the viewing window?

Answer: It may have no solution or intersect outside the window. Zoom out or check for parallel lines algebraically.

Flashcard 5: What is the solution of the system $3x+y=2$ and $6x+2y=5$?

Answer: No solution. Adding gives $9x+3y=7$, which contradicts first.

Flashcard 6: What is the solution of the system $x+3y=0$ and $2x-3y=15$?

Answer: $(5,-\frac{5}{3})$. Add equations to eliminate $y$: $3x=15$, so $x=5$.

Flashcard 7: What is the elimination goal when you add two equations in a system?

Answer: Make one variable coefficient sum to $0$. Creates single-variable equation when adding.

Flashcard 8: What is the solution of the system $7x+2y=20$ and $7x-2y=8$?

Answer: $(2,3)$. Add equations to eliminate $y$: $14x=28$, so $x=2$.

Flashcard 9: What is the solution of the system $x+y=10$ and $x-y=2$?

Answer: $(6,4)$. Add equations to get $2x=12$, so $x=6$.

Flashcard 10: What does it mean if a system of two linear equations has exactly one solution?

Answer: The lines intersect at exactly one point $(x,y)$. Two distinct lines meet at one unique point.

Flashcard 11: What is the solution of the system $y=6$ and $2x+y=10$?

Answer: $(2,6)$. Substitute $y=6$ into second equation.

Flashcard 12: What is the approximate solution if the intersection appears at about $(2.1,3.9)$ on a graph?

Answer: $(2.1,3.9)$ (approx.). Read coordinates where lines visually cross.

Flashcard 13: What is the standard form of a linear equation used for elimination?

Answer: $Ax+By=C$. Format ideal for elimination method alignment.

Flashcard 14: What method solves a system by making coefficients opposites and adding equations?

Answer: Elimination (linear combination). Add equations after making coefficients opposites.

Flashcard 15: What is the solution of the system $y=\frac{1}{2}x+3$ and $y=-\frac{1}{2}x+1$?

Answer: $(-2,2)$. Set slopes equal: $\frac{1}{2}x+3=-\frac{1}{2}x+1$, solve.

Flashcard 16: What does it mean if a system of two linear equations has infinitely many solutions?

Answer: The lines coincide: equivalent equations for the same line. Every point on the line satisfies both equations.

Flashcard 17: What is the solution of the system $0.5x+y=4$ and $x-2y=0$?

Answer: $(4,2)$. Same as previous with $0.5=\frac{1}{2}$.

Flashcard 18: What is the solution of the system $3x+y=2$ and $6x+2y=4$?

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

Flashcard 19: What is the solution of the system $x-\frac{1}{2}y=3$ and $2x-y=6$?

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

Flashcard 20: What is the solution type for $y=2x+1$ and $y=2x-3$?

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

Flashcard 21: What method solves a system by finding the intersection point of two graphed lines?

Answer: Graphing (approximate intersection). Visual method showing where lines cross.

Flashcard 22: What is the solution of the system $2x-y=1$ and $-4x+2y=-2$?

Answer: Infinitely many solutions. Second equation is first multiplied by $-2$.

Flashcard 23: What is the solution type for $3x-6y=9$ and $x-2y=5$?

Answer: No solution (parallel lines). Same slope $m=\frac{3}{2}$, different intercepts.

Flashcard 24: What is the solution of the system $2x-y=1$ and $-2x+y=5$?

Answer: No solution. Adding gives $0=6$, which is false.

Flashcard 25: What is the solution of the system $\frac{1}{2}x+y=4$ and $x-2y=0$?

Answer: $(4,2)$. Substitute second into first: $\frac{1}{2}(2y)+y=4$.

Flashcard 26: Identify the system solution type if elimination gives a false statement like $0=5$.

Answer: No solution. Variables cancel to give contradiction.

Flashcard 27: What is the first step to solve $y=mx+b$ and $Ax+By=C$ by substitution?

Answer: Substitute $y=mx+b$ into $Ax+By=C$. Replace $y$ in the second equation with the expression.

Flashcard 28: What is the solution of the system $2x+3y=0$ and $4x-3y=21$?

Answer: $(\frac{7}{2},-\frac{7}{3})$. Add equations to eliminate $y$: $6x=21$, so $x=\frac{7}{2}$.

Flashcard 29: What is the x-intercept of the line $4x+2y=8$?

Answer: $(2,0)$. Set $y=0$: $4x=8$, so $x=2$.

Flashcard 30: What is the solution of the system $4x-y=9$ and $2x+y=3$?

Answer: $(2,-1)$. Add equations to eliminate $y$: $6x=12$, so $x=2$.

Flashcard 31: What is the solution of the system $x+2y=9$ and $y=4$?

Answer: $(1,4)$. Substitute $y=4$ into first equation.

Flashcard 32: What is the solution of the system $y=2x$ and $y=2x+5$?

Answer: No solution. Parallel lines with same slope, different intercepts.

Flashcard 33: What is the solution type for $2x+4y=8$ and $x+2y=4$?

Answer: Infinitely many solutions (equivalent equations). Second equation is first divided by 2.

Flashcard 34: What is the solution of the system $y=-3x+6$ and $3x+y=6$?

Answer: Infinitely many solutions. Second equation rearranges to first equation form.

Flashcard 35: Identify the system solution type if elimination gives a true statement like $0=0$.

Answer: Infinitely many solutions. Variables cancel to give identity $0=0$.

Flashcard 36: Identify the condition for the same line in $y=mx+b$ form.

Answer: Equal slopes and intercepts: $m_1=m_2$ and $b_1=b_2$. Identical lines when both parameters match.

Flashcard 37: What is the solution of the system $y=2x+1$ and $y=x+4$?

Answer: $(3,7)$. Set equations equal: $2x+1=x+4$, solve for $x=3$.

Flashcard 38: What is the solution of the system $y=x$ and $y=-x$?

Answer: $(0,0)$. Lines $y=x$ and $y=-x$ intersect at origin.

Flashcard 39: What is the solution of the system $y=-x+2$ and $y=x-4$?

Answer: $(3,-1)$. Set equations equal: $-x+2=x-4$, solve for $x=3$.

Flashcard 40: What operation keeps a linear equation equivalent when preparing for elimination?

Answer: Multiply or divide both sides by the same nonzero constant. Preserves solution set for elimination setup.

Flashcard 41: Identify the condition for parallel lines in $y=mx+b$ form.

Answer: Equal slopes $m_1=m_2$ and different intercepts $b_1\ne b_2$. Lines never meet when slopes match but intercepts differ.

Flashcard 42: What is the solution of the system $5x+y=1$ and $x-y=7$?

Answer: $(\frac{4}{3},-\frac{17}{3})$. Add equations to eliminate $y$: $6x=8$, so $x=\frac{4}{3}$.

Flashcard 43: What is the solution of the system $y=3x-5$ and $2y=6x-10$?

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

Flashcard 44: What is the solution of the system $x+y=5$ and $2x+2y=8$?

Answer: No solution. Second equation gives $x+y=4$, contradicting first.

Flashcard 45: What is the slope-intercept form of a linear equation?

Answer: $y=mx+b$. Standard form showing slope $m$ and $y$-intercept $b$.

Flashcard 46: What method solves a system by solving one equation for a variable and substituting?

Answer: Substitution. Replace one variable with expression from other equation.

Flashcard 47: What is the solution of the system $x=4$ and $y=-2x+1$?

Answer: $(4,-7)$. Substitute $x=4$ into second equation.

Flashcard 48: What is the solution of the system $3x-2y=4$ and $x+2y=8$?

Answer: $(3,\frac{5}{2})$. Add equations to eliminate $y$: $4x=12$, so $x=3$.

Flashcard 49: What does it mean if a system of two linear equations has no solution?

Answer: The lines are parallel: same slope, different $y$-intercepts. Never intersect since they have identical slopes.

Flashcard 50: What is the $y$-intercept of the line $2x+5y=10$?

Answer: $(0,2)$. Set $x=0$: $5y=10$, so $y=2$.

Flashcard 51: Identify the best method for $y=3x+2$ and $2x+y=9$ if you want minimal rewriting.

Answer: Substitution. First equation already solved for $y$.

Flashcard 52: What is the slope of the line $3x-6y=12$?

Answer: $\frac{1}{2}$. Rewrite as $y=\frac{1}{2}x-2$ to find slope.