Question

Solve the system of linear equations: 1/5 x + 1/8 y = 1 1/2 x − 1/3 y = 1

Answer 1

`x = (110)/(31); \qquad y = (72 )/(31)`

Explanation:

I am guessing that your two equations are

(1) ⅕x + ⅛y = 1

(2) ½x - ⅓ y = 1

To get rid of fractions, I would multiply each equation by the least common multiple of its denominators.

`\begin{array}{rcrl}(3) \qquad 8x + 5y & = & 40 & \text{Multiplied (1) by 40}\\(4) \qquad 3x - 2y & = & 6 & \text{Multiplied (2) by 6}\\\end{array}`

We can solve this system of equations by the method of elimination.

`\begin{array}{rcrl}(5) \qquad \, \, 16x + 10y & = & 80 & \text{Multiplied (3) by 2}\\(6) \qquad \, \: 15x - 10 y & = & 30 & \text{Multiplied (4) by 5}\\31x & = & 110 & \text{Added (5) and (6)}\\\\(7)\qquad\qquad \qquad x & = & (110 )/(31) & \text{Divided each side by 31}\\\end{array}`

`\begin{array}{rcrl}3 \left ((110)/(31) \right) - 2y & = & 6 & \text{Substituted (7) into (4)}\\\\ (5) \qquad16x + 10y & = & 80 & \text{Multiplied (3) by 2}\\\\(6)\qquad 15x - 10 y & = & 30 & \text{Multiplied (4) by 5}\\\\31x & = & 110 & \text{Added (5) and (6)}\\\\(7)\qquad \qquad \qquad x & = & (110 )/(31) & \text{Divided each side by 31}\\\\3 \left((110)/(31) \right ) - 2y & = & 6 & \text{Substituted (7) into (4)}\\\\\end{array}\\\\`

`\begin{array}{rcll}(330)/(31) - 2y & = & 6 &\\\\-2y & = & 6 - (330)/(31) &\\\\y & = & (165)/(31) -3 & \text{Divided each side by -2}\\\\ & = & (165 - 93)/(31) &\\\\ & = & (72)/(31) &\\\\\end{array}\\\\\therefore x = (110)/(31); \qquad y = (72 )/(31)`

The diagram below shows the graphs of your two functions intersecting at (3.548, 2.323). These are the decimal equivalents of your fractional coordinates.

Answer 2

x = 75 and y = -72

Explanation:

It is given that,

1/5 x + 1/8 y = 1 ------(1)

1/2 x − 1/3 y = 1 -------(2)

To find the solutions of the system of equations

Step 1: eq(1) * 5 ⇒

x + 5/8y = 5 ----(3)

Step 2: eq(2) * 2 ⇒

x - 2/3y = 2 -----(4)

Step 3: eq(3) - eq(4) ⇒

x + 5/8y = 5 ----(3)

x - 2/3y = 2 -----(4)

0 +(5/8 - 2/3)y = 3

-1/24 y = 3

y = -24*3 = -72

Step 4: Substitute the value of y in eq(1)

1/5 x + 1/8 y = 1 ------(1)

1/5 x + 1/8 (-72) = 1 ------(1)

1/5 x - 24 = 1

1/5 x = 25

x = 5*25 = 75

Therefor x = 75 and y = -72