Math Problem Statement

Solve the systems of equations: a) y + x/5 = 1; 3y + x/5 = 14 b) x + y/3 = 9; x/5 - y/2 + 1/2 = 0 c) x/5 = 3/4; x/4 = 2 - 1 d) x + 1/4 = 2y - 8; (x + y)/2 - (x - y)/3 = 1 e) 1/x + 1/y = 1; 1/x - 1/y = 9 f) 2/x + 1/y = 5; 1/x + 3/y = 5

Solution

Here are the given systems of equations from the image:

a)
y+x5=1y + \frac{x}{5} = 1 3y+x5=143y + \frac{x}{5} = 14

b)
x+y3=9x + \frac{y}{3} = 9 x5y2+12=0\frac{x}{5} - \frac{y}{2} + \frac{1}{2} = 0

c)
x5=34\frac{x}{5} = \frac{3}{4} x4=21\frac{x}{4} = 2 - 1

d)
x+14=2y8x + \frac{1}{4} = 2y - 8 x+y2xy3=1\frac{x + y}{2} - \frac{x - y}{3} = 1

e)
1x+1y=1\frac{1}{x} + \frac{1}{y} = 1 1x1y=9\frac{1}{x} - \frac{1}{y} = 9

f)
2x+1y=5\frac{2}{x} + \frac{1}{y} = 5 1x+3y=5\frac{1}{x} + \frac{3}{y} = 5

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Math Problem Analysis

Mathematical Concepts

Systems of Equations
Linear Equations
Fractional Equations

Formulas

Addition and Subtraction of Fractions
Multiplying/Dividing Fractions
Solving Systems by Substitution/Elimination

Theorems

Linear Systems Solution Theorem
Fractional Equation Properties

Suitable Grade Level

Grades 9-11