Question

Graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. (x − 2)²+(y+3)² = 4, y = x - 3

Accepted Answer

Identify the two equations given: the first is a circle equation $$\left(x - 2\$$right)^2$$ + \left(y + 3\$$right)^2$$ = 4$$, and the second is a line equation $$y = x - 3. $$Substitute the expression for $$y$$ from the line equation into the circle equation. Replace $$y$$ with $$x - 3 in$$ $$\left(x - 2\$$right)^2$$ + \left(y + 3\$$right)^2$$ = 4 to $$get $$\left(x - 2\$$right)^2$$ + \left((x - 3) + 3\$$right)^2$$ = 4. $$Simplify the equation after substitution: note that $$((x - 3) + 3)$$ simplifies to $$x$$, so the equation becomes $$\left(x - 2\$$right)^2$$ + x^2$ = 4. $$Expand the squared terms and combine like terms to form a quadratic equation in terms of $$x. $$That is, expand $$\left(x - 2\$$right)^2$$ to x^2$ - 4x + 4$$ and add $$x^2$$, resulting in $$x^2 - 4x + 4$$ + $$x^2 = 4$. Solve the quadratic equation for x$, then substitute each solution back into y = x - 3$ to find the corresponding y$ values. These (x$$, y)$$ pairs are the points of intersection. Finally, verify each point by plugging them into both original equations to confirm they satisfy both.

Graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. (x − 2)²+(y+3)² = 4, y = x - 3

Risposta video verificata per un problema simile:

Concetti chiave

Graphing Circles

Graphing Linear Equations

Finding Points of Intersection