What is solution of simultaneous equations `7^x-3y=43`  ;  `4y+2*7^x=106` ?

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lemjay | High School Teacher | (Level 3) Senior Educator

Posted on

Let,

EQ.1: `7^x-3y=43`

EQ.2: `4y+2*7^x=106`

To solve for x and y, use elimination method.

> Multiply EQ.1 by (-2), the add it with EQ.2.

`-2(7^x - 3y) = 43(-2` )    ==>`-2*7^x + 6y = -86`                   

                                             `(+)`   `2*7^x + 4y = 106`

                                            ----------------------------

                                                              `10y = 20`

                                                                 ` y = 2`

> Then, substitute value of y to either EQ.1 or EQ.2.

`7^x - 3y = 43`

`7^x - 3(2) = 43`

 ` 7^x - 6 = 43`

         ` 7^x = 49`

> Express 49 with its prime factor. (Note 49 = 7*7 = 7^2)

         ` 7^x = 7^2`

> Since both sides of the equation have the same base, equate the exponent of the left side equal to exponent of right side.

             `x = 2`

Answer: x=2 and y=2

sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You need to use the first equation to write `7^x`  in terms of y such that:

`7^x = 3y + 43`

You need to susbtitute `3y + 43`  for `7^x`  in the next equation such that:

`4y+2*(3y + 43)=106`

You need to solve for y the equation such that:

`4y + 6y + 86 - 106 = 0`

`10y - 20 = 0 =gt 10y = 20 =gt y = 2`

You need to substitute 2 for y in equation `7^x = 3y + 43`  such that:

`7^x = 6 + 43 =gt 7^x = 49 =gt 7^x = 7^2 =gt x = 2`

Hence, evaluating the solution to the system of equations yields `x = 2 ; y = 2` .

Sources:

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