# Systems of Equations - Problems & Answers

Problem 1 Two of the following systems of equations have solution (1;3). Find them out by checking.
a) $\begin{array}{|l} x + y = 5 \\ 2x - y = 7; \end{array}$

b) $\begin{array}{|l} 2x + y = 5 \\ x - y = 2 \end{array}$

c) $\begin{array}{|l} 3x + y = 6 \\ 4x - 3y = -5; \end{array}$

d) $\begin{array}{|l} \frac{1}{x - 1} = y-3 \\ x - y = -2; \end{array}$

e) $\begin{array}{|l} \frac{9x + 4y}{3}-\frac{5x-11}{2} = 13-y \\ 13x - 7y = -8; \end{array}$

Answer: c and e has solution (1; 3)

Problem 2 Are the systems equivalent(check if solutions of the both systems are the same):
$\begin{array}{|l} 4x + 5y = 11 \\ x - y = 5 \end{array}$
and
$\begin{array}{|l} 4x - 5y = 11 \\ 2x + y = 9? \end{array}$

(3-32) Solve the system of equations:

Problem 3
$\begin{array}{|l} 2x - y = -5 \\ y = 1-3x \end{array}$
Answer: x = 1, y = -2.

Problem 4
$\begin{array}{|l} 3x - y = 13 \\ 3y - 2x = -4 \end{array}$
Answer: x = 5, y = 2.

Problem 5
$\begin{array}{|l} 6x - y = 11 \\ 12x - 2y - 22 = 0 \end{array}$
Answer: The solution is every couple of numbers, which is solution of the equation $6x - y = 11$.

Problem 6
$\begin{array}{|l} 5u - 6v = -2 \\ 7u + 18v = 2 \end{array}$
Answer: $x = -1, y = \frac{1}{2}$.

Problem 7
$\begin{array}{|l} 8x - 5y + 16 = 0 \\ 1x + 3y - 17 = 0 \end{array}$
Answer: $x = \frac{1}{2}, y = 4$.

Problem 8
$\begin{array}{|l} 4(x + 2) - 7(x - y) = 7 \\ 7(x + y) + 10(x - 2) = 79 \end{array}$
Answer: $x = 5, y = 2.$

Problem 9
$\begin{array}{|l} 3x + 4(x - 3) = 3(2y - 3) - 3y \\ 3y + 2(x - 4) = 5(y + 2) - 28\end{array}$

Problem 10
$\begin{array}{|l} (x + 3)(x-1) = 4y + x^2 + 5 \\ (x - 3)(3x + 2) = 3x^2 - 14y + 15\end{array}$

Problem 11
$\begin{array}{|l} (x - 1)(y + 2) - (x - 2)(y + 5) = 0 \\ (x + 4)(y - 3) - (x + 7)(y - 4) = 0\end{array}$
Answer: x = 5, y = 7.

Problem 12
$\begin{array}{|l} (x + 2)^2 - (x + 3)(x - 3) - 3(y + 5) = 0 \\ (2y - 3)^2 - y(4y - 3) + 12x - 15 = 0\end{array}$
Answer: The solution is every couple, which is solution to the equation 4x - 3y - 2 = 0.

Problem 13
$\begin{array}{|l} \frac{y + 2}{6} - \frac{y-4}{2} = \frac{x}{3} \\ \frac{4}{3}(y - 1) - 2x = -2\end{array}$
Answer: x = 3, y = 4.

Problem 14
$\begin{array}{|l} 0.25x - 0.04y = 1 \\ 0.4x + 1.5y = 40.7\end{array}$
Answer: x = 8, y = 25.

Problem 15
$\begin{array}{|l} \frac{5x-3y}{4} = \frac{x-5y}{3} \\ 7x + y = 12\end{array}$
Answer: x = 2, y = -2

Problem 16
$\begin{array}{|l} \frac{3x+1}{5}+2y-3 = 0 \\ \frac{4y-5}{6}+3y-9 = -\frac{1}{2}\end{array}$
Answer: $x = -\frac{42}{11}, y = \frac{28}{11}$

Problem 17
$\begin{array}{|l} \frac{3x-1}{5}+3y-4 = 15 \\ \frac{3y-5}{6}+2x-8 = \frac{23}{3}\end{array}$
Answer: $x = 7, y = 5$

Problem 18
$\begin{array}{|l} \frac{2x-z}{6}+\frac{2x-z}{9} = 3 \\ \frac{x+z}{3}-\frac{x-z}{4} = 4\end{array}$
Answer: x = y = 6

Problem 19
$\begin{array}{|l} \frac{x-1}{3} + \frac{5y+1}{2} = \frac{x+10y-8}{6} \\ \frac{(x+2)(5y-2)}{2} = 5+\frac{5xy}{2}-2(x+1)\end{array}$

Problem 20
$\begin{array}{|l} \frac{5x-1}{6} + \frac{3y-1}{10} = 3 \\ \frac{11-x}{6} + \frac{11+y}{4} = 3\end{array}$
Answer: x = 5, y = -3.

Problem 21
$\begin{array}{|l}y-0.2(x - 2) = 1.4\\ \frac{5}{2} - \frac{2y - 3}{4} = \frac{4x - y}{8}\end{array}$
Answer: $x = 5, y = 2.$

Problem 22
$\begin{array}{|l}\frac{x}{5} + 0.03(10y - 20) = 0.8\\ \frac{2x + 4.5}{20} - 0.75 = \frac{y - 3}{8}\end{array}$
Answer: $x = 4, y = 2.$

Problem 23
$\begin{array}{|l}y-x-\frac{5x-4}{2}=3-\frac{11y+17}{4}\\ x+\frac{9y+11}{4}-\frac{3y+4}{7}=6\end{array}$
Answer: $x = 2, y = 1.$

Problem 24
$\begin{array}{|l}\frac{5x-3y}{3}-\frac{2y-3x}{5}=x+1\\ \frac{2x-3y}{3}-\frac{3y-4x}{2}=y+1\end{array}$
Answer: $x = 3, y = 2.$

Problem 25
$\begin{array}{|l}\frac{x-1}{4}\frac{1+y}{2}=\frac{1}{6}-\frac{x+2y}{6}\\ \frac{x-2}{3}+\frac{x}{15}=\frac{y+4}{5}-\frac{4x-y}{15}\end{array}$

Answer: The solution is every couple, which is solution of the equation $5x - 2y = 11.$

Problem 26
$\begin{array}{|l}\frac{x+2y}{4}-\frac{x-2y}{2}=1-\left[x-\frac{7-2y}{3}\right]\\ 3x-2y=8\end{array}$
Answer: $x = 3, y = \frac{1}{2}.$

Problem 27
$\begin{array}{|l}\frac{7+x}{5}-\frac{2x-y}{4}-3y=-5\end{array}$
$\begin{array}{|l}\frac{5y-7}{2}+\frac{4x-3}{6}-18=-5x\end{array}$
Answer: x = 3, y = 2.

Problem 28
$\begin{array}{|l}\frac{11y}{20}-0.8\left(\frac{x}{4}+2.5\right)=\frac{5}{2}\end{array}$
$\begin{array}{|l}\frac{6x-0.3y}{2}-\frac{3}{2}=2(1+x)\end{array}$
Answer: x = 5, y = 10.

Problem 29
$\begin{array}{|l}0.5x-\frac{y-4}{5}=0.3x-\frac{y-4}{2}\end{array}$
$\begin{array}{|l}0.5y-\frac{x-4}{6}=\frac{7y}{12}-\frac{x-3}{3}\end{array}$
Answer: x = 3, y = 2.

Problem 30
$\begin{array}{|l}\frac{2(x-y)}{3}+1.6=\frac{8x}{15}-\frac{3y-10}{5}\end{array}$
$\begin{array}{|l}\frac{3x+4}{4}+\frac{y}{8}=\frac{5x}{6}-\frac{y-17}{12}\end{array}$
Answer: x = 5, y = 4.

Problem 31
$\begin{array}{|l}\frac{(2+x)(5y-2)}{2}=5+\frac{5xy}{2}-2(1+x)\end{array}$
$\begin{array}{|l}(x-1)^2+(2y+1)^2=2(1+2y)(x-1)\end{array}$

Answer: The solution is every couple, which is solution of the equation x + 5y = 5

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