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{:[y=(1)/(3)x+5],[y=2x]:} Consider the given system of equations. If (x,y) is the solution to the system, then what is the value of y+x ?

y=13x+5y=2x \begin{array}{l} y=\frac{1}{3} x+5 \\ y=2 x \end{array} \newlineConsider the given system of equations. If (x,y) (x, y) is the solution to the system, then what is the value of y+x y+x ?

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Q. y=13x+5y=2x \begin{array}{l} y=\frac{1}{3} x+5 \\ y=2 x \end{array} \newlineConsider the given system of equations. If (x,y) (x, y) is the solution to the system, then what is the value of y+x y+x ?
  1. Write System of Equations: Write down the system of equations.\newlineWe have the following system of equations:\newliney=13x+5 y = \frac{1}{3}x + 5 \newliney=2x y = 2x \newlineWe need to find the values of x and y that satisfy both equations simultaneously.
  2. Set Equations Equal: Set the two equations equal to each other to solve for x.\newlineSince both expressions are equal to y, we can set them equal to each other:\newline13x+5=2x \frac{1}{3}x + 5 = 2x
  3. Subtract to Isolate x: Subtract 13x\frac{1}{3}x from both sides to start isolating x.\newline5=2x−13x 5 = 2x - \frac{1}{3}x
  4. Combine Like Terms: Combine like terms on the right side.\newlineTo combine the terms, we need a common denominator. The common denominator for 22 and 13\frac{1}{3} is 33, so we convert 22x to 63x\frac{6}{3}x:\newline5=63x−13x 5 = \frac{6}{3}x - \frac{1}{3}x \newline5=53x 5 = \frac{5}{3}x
  5. Multiply by Reciprocal: Multiply both sides by the reciprocal of 53\frac{5}{3} to solve for x.\newlinex=5⋅35 x = 5 \cdot \frac{3}{5} \newlinex=3 x = 3
  6. Substitute x Value: Substitute x = 33 into one of the original equations to solve for y.\newlineWe can use the second equation y=2x y = 2x for simplicity:\newliney=2⋅3 y = 2 \cdot 3 \newliney=6 y = 6
  7. Add x and y Values: Add the values of x and y to find y + x.\newliney+x=6+3 y + x = 6 + 3 \newliney+x=9 y + x = 9

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