Ch. 5 - Systems and Matrices

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 5 - Systems and Matrices

Problem 64

Chapter 6, Problem 64

Solve each problem. Find the equation of the line passing through the points of intersection of the graphs of x² + y² = 20 and x² - y = 0.

Verified step by step guidance

Identify the two given equations: the circle $x^2 + y^2 = 20$ and the parabola $x^2 - y = 0$.

Express $y$ from the parabola equation: $y = x^2$.

Substitute $y = x^2$ into the circle equation to find the points of intersection: $x^2 + (x^2)^2 = 20$, which simplifies to $x^2 + x^4 = 20$.

Solve the resulting equation for $x$ to find the $x$-coordinates of the intersection points, then use $y = x^2$ to find the corresponding $y$-coordinates.

Use the two points of intersection to find the equation of the line passing through them by calculating the slope $m = \frac{y_2 - y_1}{x_2 - x_1}$ and then using the point-slope form $y - y_1 = m(x - x_1)$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Finding Points of Intersection

To find where two graphs intersect, solve their equations simultaneously. This involves substituting one equation into the other or using algebraic methods to find common solutions (x, y) that satisfy both equations.

Equation of a Line Through Two Points

Once two points are known, the equation of the line passing through them can be found using the slope formula and point-slope form. The slope is the change in y over the change in x, and the line equation can be expressed in forms like y = mx + b.

Substitution Method in Systems of Equations

The substitution method involves solving one equation for one variable and substituting that expression into the other equation. This simplifies the system to a single-variable equation, making it easier to find the intersection points.

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Related Practice

Textbook Question

Answer each question. A line passes through the points of intersection of the graphs of y = x² and x² + y² = 90. What is the equation of this line?

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Textbook Question

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7.

4x + 3y = -7

2x + 3y = -11

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Textbook Question

Graph the solution set of each system of inequalities.

$y \leq \log x$

$y \geq ∣ x - 2 ∣$

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Textbook Question

Answer each question. Does the straight line 3x - 2y = 9 intersect the circle x² + y² = 25? (Hint: To find out, solve the system formed by these two equations.)

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Textbook Question

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7.

x + y = 4

2x - y = 2

919

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Textbook Question

Graph the solution set of each system of inequalities.

$y$\ge$3^{x}$