Which system is equivalent to $\left\{\begin{array}{r}y=9 x^2 \\ x+y=5\end{array}\right.$ ?
$\left\{\begin{aligned} 5-x & =9 x^2 \\ y & =5-x\end{aligned}\right.$
$\left\{\begin{array}{l}y=9 y^2-90 y+225 \\ x=y-5\end{array}\right.$
$\left\{\begin{aligned} 5+x & =9 x^2 \\ y & =5+x\end{aligned}\right.$
$\left\{\begin{aligned} y & =3 x \\ x+3 x & =5\end{aligned}\right.$
Answer (1)
Express y in terms of x from the second equation: y = 5 − x .
Substitute this expression into the first equation: 5 − x = 9 x 2 .
The equivalent system is then { 5 − x = 9 x 2 y = 5 − x .
The final answer is { 5 − x = 9 x 2 y = 5 − x .
Explanation
Analyze the problem We are given the system of equations:
{ y = 9 x 2 x + y = 5
Our goal is to find an equivalent system of equations from the given options.
Find the equivalent system From the second equation, we can express y in terms of x :
y = 5 − x
Now, substitute this expression for y into the first equation:
5 − x = 9 x 2
So, the equivalent system is:
{ 5 − x = 9 x 2 y = 5 − x
Identify the correct option Comparing this with the given options, we see that the first option matches our result:
{ 5 − x = 9 x 2 y = 5 − x
State the final answer Therefore, the equivalent system is:
{ 5 − x = 9 x 2 y = 5 − x
Examples
Understanding equivalent systems of equations is crucial in various fields, such as physics and engineering, where multiple equations describe a single phenomenon. For example, in circuit analysis, you might have two equations describing the voltage and current relationships in a circuit. Finding an equivalent system can simplify the analysis and help you solve for unknown variables more easily. This skill is also fundamental in economics, where you might analyze supply and demand curves to find equilibrium points.
