2. Illustrate or graph the system of non-linear equations then find its solution/s.a) y = x² y = √xb) y-x² = 0 y-x²= 2 =Systems of Non-linear Equation Pre-Calculu​s

Answer:**a) $y = x^2$ and $y = \sqrt{x}$**AnalysisThis system involves a parabola ($y = x^2$) and a square root function ($y = \sqrt{x}$). We'll solve it algebraically and then illustrate the solution graphically. Note that the square root function is only defined for non-negative x values.Step 1 Set the equations equal to each other since both are equal to $y$:$x^2 = \sqrt{x}$Step 2 Square both sides to eliminate the square root:$(x^2)^2 = (\sqrt{x})^2$$x^4 = x$Step 3 Rearrange the equation into a standard polynomial form:$x^4 - x = 0$$x(x^3 - 1) = 0$Step 4 Solve for $x$:This gives us two solutions for $x$: $x = 0$ and $x^3 = 1$, which means $x = 1$.Step 5 Find the corresponding $y$ values:For $x = 0$, $y = 0^2 = 0$ or $y = \sqrt{0} = 0$.For $x = 1$, $y = 1^2 = 1$ or $y = \sqrt{1} = 1$.Step 6 State the solutions:The solutions are $(0, 0)$ and $(1, 1)$.Step 7 Graphical illustration:The graph would show a parabola ($y = x^2$) opening upwards and a square root function ($y = \sqrt{x}$) starting at the origin and increasing gradually. The two curves intersect at (0,0) and (1,1).AnswerThe solutions to the system of equations $y = x^2$ and $y = \sqrt{x}$ are $(0, 0)$ and $(1, 1)$.**b) $y - x^2 = 0$ and $y - x^2 = 2$**AnalysisThis system involves two parabolas that are parallel to each other. The second equation is simply the first equation shifted vertically upwards by 2 units. Let's solve it algebraically.Step 1 Rewrite the equations:$y = x^2$$y = x^2 + 2$Step 2 Notice that there is no solution.The first equation states that $y$ is always equal to $x^2$. The second equation states that $y$ is always 2 units greater than $x^2$. There is no value of $x$ that can satisfy both equations simultaneously. The parabolas are parallel and do not intersect.Step 3 Graphical illustration:The graph would show two parabolas, both opening upwards. One parabola ($y = x^2$) is positioned at the origin, and the other ($y = x^2 + 2$) is shifted two units vertically above the first. They do not intersect.AnswerThere are no solutions to the system of equations $y - x^2 = 0$ and $y - x^2 = 2$. The two equations represent parallel parabolas that do not intersect.Step-by-step explanation:sana nakatulong!

Answered by alforquejhelaiza | 2025-07-29