{:[y=(1)/(5)x-9],[5y^(2)+15=7x]:} If (a,b) is a solution to the system of equations shown and b 0, what is the value of b ?

Question

{:[y=(1)/(5)x-9],[5y^(2)+15=7x]:} If  (a,b) is a solution to the system of equations shown and  b > 0, what is the value of  b ?

Bytelearn · Accepted Answer

Write Equations: Write down the system of equations. We have the following system of equations: $$ y = \frac{1}{5}x - 9 $$ $$ 5y^2 + 15 = 7x $$Express x in terms: Express x in terms of y from the first equation. From the first equation, we can solve for x: $$ y = \frac{1}{5}x - 9 $$ $$ y + 9 = \frac{1}{5}x $$ $$ x = 5(y + 9) $$Substitute x into second: Substitute the expression for x from Step 2 into the second equation. We substitute $ x = 5(y + 9) $ into the second equation: $$ 5y^2 + 15 = 7x $$ $$ 5y^2 + 15 = 7(5(y + 9)) $$Simplify and solve for y: Simplify the equation and solve for y. $$ 5y^2 + 15 = 35(y + 9) $$ $$ 5y^2 + 15 = 35y + 315 $$ $$ 5y^2 - 35y + 15 - 315 = 0 $$ $$ 5y^2 - 35y - 300 = 0 $$Factor quadratic equation: Factor the quadratic equation. We need to factor the quadratic equation $ 5y^2 - 35y - 300 = 0 $. This can be done by looking for two numbers that multiply to $ 5 \times -300 = -1500 $ and add up to -35. These numbers are -60 and +25. $$ 5y^2 - 60y + 25y - 300 = 0 $$ $$ 5y(y - 12) + 25(y - 12) = 0 $$ $$ (5y + 25)(y - 12) = 0 $$Solve for y: Solve for y. Setting each factor equal to zero gives us two possible solutions for y: $$ 5y + 25 = 0 $$ $$ y = -5 $$ and $$ y - 12 = 0 $$ $$ y = 12 $$Choose positive y value: Choose the value of y that satisfies the condition b > 0. Since we are given that b > 0 and b corresponds to y in our system of equations, we choose the positive value of y: $$ y = 12 $$

$\begin{aligned} y & =\frac{1}{5} x-9 \\ 5 y^{2}+15 & =7 x \end{aligned}$ $\newline$ If $(a, b)$ is a solution to the system of equations shown and b>0 , what is the value of $b$ ?

Full solution

More problems from Multiply two decimals: where does the decimal point go?

Related topics

yamp;=15x−95y2+15amp;=7x \begin{aligned} y &amp; =\frac{1}{5} x-9 \\ 5 y^{2}+15 &amp; =7 x \end{aligned} y5y2+15​amp;=51​x−9amp;=7x​\newlineIf (a,b) (a, b) (a,b) is a solution to the system of equations shown and b>0 , what is the value of b b b ?

$\begin{aligned} y & =\frac{1}{5} x-9 \\ 5 y^{2}+15 & =7 x \end{aligned}$ $\newline$ If $(a, b)$ is a solution to the system of equations shown and b>0 , what is the value of $b$ ?