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x+6y=r-2x 2(x+4)+2y=11+(1)/(2)(16+2x) In the system of equations, r is a constant. For what value of r does the system of linear equations have infinitely many solutions?

x+6y=r−2x x+6 y=r-2 x \newline2(x+4)+2y=11+12(16+2x) 2(x+4)+2 y=11+\frac{1}{2}(16+2 x) \newlineIn the system of equations, r r is a constant. For what value of r r does the system of linear equations have infinitely many solutions?

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Q. x+6y=r−2x x+6 y=r-2 x \newline2(x+4)+2y=11+12(16+2x) 2(x+4)+2 y=11+\frac{1}{2}(16+2 x) \newlineIn the system of equations, r r is a constant. For what value of r r does the system of linear equations have infinitely many solutions?
  1. Substitute and Simplify: Substitute xx from the first equation into the second equation to express everything in terms of yy and rr.x+6y=r−2xx + 6y = r - 2x⇒3x+6y=r\Rightarrow 3x + 6y = rNow, double the first equation to get a comparable term for substitution.2(x+6y)=2(r−2x)2(x + 6y) = 2(r - 2x)⇒2x+12y=2r−4x\Rightarrow 2x + 12y = 2r - 4x⇒6x+12y=2r\Rightarrow 6x + 12y = 2r
  2. Double Equations: Now let's simplify the second given equation.\newline2(x+4)+2y=11+(12)(16+2x)2(x + 4) + 2y = 11 + \left(\frac{1}{2}\right)(16 + 2x)\newline⇒2x+8+2y=11+8+x\Rightarrow 2x + 8 + 2y = 11 + 8 + x\newline⇒2x+2y=11+x\Rightarrow 2x + 2y = 11 + x\newline⇒x+2y=11\Rightarrow x + 2y = 11
  3. Simplify Second Equation: Double the modified second equation to compare it with the modified first equation.\newline2(x+2y)=2(11)2(x + 2y) = 2(11)\newline⇒2x+4y=22\Rightarrow 2x + 4y = 22\newlineNow, multiply this by 33 to match the terms with the first modified equation.\newline3(2x+4y)=3(22)3(2x + 4y) = 3(22)\newline⇒6x+12y=66\Rightarrow 6x + 12y = 66
  4. Compare and Multiply: Compare the two modified equations.\newline6x+12y=2r6x + 12y = 2r from the first modification\newline6x+12y=666x + 12y = 66 from the second modification\newlineFor the system to have infinitely many solutions, the two equations must be identical.\newlineSo, set 2r2r equal to 6666.
  5. Set Equal and Solve: Solve for rr.r=662r = \frac{66}{2}r=33r = 33

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