Algebra

QUESTION 1 [2 marks] Easy

Solve the system: $x+y=10$ and $x-y=2$.

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Add the two equations: $2x=12 \Rightarrow x=6$. Substitute into $x+y=10$: $y=4$. So $x=6,\ y=4$.

QUESTION 2 [2 marks] Easy

Solve the system: $2x+3y=12$ and $x=3$.

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Substitute $x=3$: $2(3)+3y=12 \Rightarrow 6+3y=12 \Rightarrow 3y=6 \Rightarrow y=2$. So $x=3,\ y=2$.

QUESTION 3 [3 marks] Medium

Solve the system: $3x+2y=16$ and $5x-2y=8$.

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Add the equations to eliminate $y$: $8x=24 \Rightarrow x=3$. Substitute into $3x+2y=16$: $9+2y=16 \Rightarrow 2y=7 \Rightarrow y=3.5$.

QUESTION 4 [3 marks] Medium

For what value of $k$ does the system $2x+ky=8$ and $4x+6y=16$ have infinitely many solutions?

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Multiply the first equation by $2$: $4x+2ky=16$. Comparing to $4x+6y=16$, the equations describe the same line exactly when $2k=6$, so $k=3$.

QUESTION 5 [4 marks] Hard

The system $x+2y=7$ and $3x-y=7$ has a solution $(x,y)$. Find the value of $x+y$.

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From the first equation, $x=7-2y$. Substitute into the second: $3(7-2y)-y=7 \Rightarrow 21-6y-y=7 \Rightarrow 21-7y=7 \Rightarrow 7y=14 \Rightarrow y=2$. Then $x=7-2(2)=3$. So $x+y=3+2=5$.

QUESTION 6 [5 marks] Hard

For which value of $c$ does the system $4x-6y=10$ and $6x-9y=c$ have infinitely many solutions?

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Multiply the first equation by $1.5$: $1.5(4x-6y)=1.5(10) \Rightarrow 6x-9y=15$. For the second equation to describe the exact same line, $c=15$.

QUESTION 7 [5 marks] Hard

A total of 150 tickets were sold for a concert, some at \$15 and the rest at \$25, for a total revenue of \$2900. How many \$25 tickets were sold?

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Let $x$ be the number of \$25 tickets, so $150-x$ tickets sold for \$15. Then $25x+15(150-x)=2900 \Rightarrow 25x+2250-15x=2900 \Rightarrow 10x=650 \Rightarrow x=65$. (Check: $25(65)+15(85)=1625+1275=2900$.)

Systems of two linear equations in two variables 7 questions