Problem 40 - Contemporary Precalculus

Question

Find the values of \(c\) and \(d\) for which both given points lie on the given straight line. $$c x+d y=-6 ; \quad(1,3) and (-2,12)$$

Step-by-Step Solution

Answer

Question: Determine the values of c and d such that the points (1, 3) and (-2, 12) lie on the line cx + dy = -6. Answer: c = -3, d = -1

Step 1: Plug in the first point (1, 3) into the equation

Substitute \(x=1\) and \(y=3\) in the equation \(cx + dy = -6\): $$c(1) + d(3) = -6$$ This simplifies to: $$c + 3d = -6$$

Step 2: Plug in the second point (-2, 12) into the equation

Substitute \(x=-2\) and \(y=12\) in the equation \(cx + dy = -6\): $$c(-2) + d(12) = -6$$ This simplifies to: $$-2c + 12d = -6$$

Step 3: Solve the system of linear equations

Now we have two equations with two unknowns: 1. \(c + 3d = -6\) 2. \(-2c + 12d = -6\) First, we can multiply equation 1 by 2 to eliminate the \(c\) term when we add the equations: $$2(c + 3d) = 2(-6)$$ $$2c + 6d = -12$$ Now, add the new equation to equation 2: $$(2c + 6d) + (-2c + 12d) = (-12) + (-6)$$ $$18d = -18$$ Finally, we can solve for \(d\) by dividing by 18: $$d = -1$$

Step 4: Find the value of c

Substitute \(d=-1\) back into either equation 1 or 2 to solve for \(c\). We'll choose equation 1: $$c + 3(-1) = -6$$ $$c - 3 = -6$$ Now, we can add 3 to both sides of the equation to solve for \(c\): $$c = -3$$ The values of \(c\) and \(d\) are \(c = -3\) and \(d = -1\), such that both given points lie on the straight line represented by \(cx + dy = -6\).

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with a common set of variables. In this exercise, we are dealing with a pair of linear equations involving the variables \(c\) and \(d\). Both equations are formed by substituting given points into the line equation \(cx + dy = -6\).
Solving systems like this involves finding values for \(c\) and \(d\) that satisfy both equations simultaneously.Here’s a quick approach to work with systems of equations:

• Write out each equation separately using the given data or values.
• Look for opportunities to add, subtract, multiply, or divide to eliminate one variable.
• Solve the remaining equation for the other variable.
• Substitute back to find the removed variable’s value.

By solving systems of equations, we effectively determine the intersection point, or solutions, that satisfy all the given conditions.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves the study of geometry using a coordinate system. Points, lines, and curves are defined using equations and coordinates.

In this problem, we use coordinate geometry to determine which line passes through both given points. The given line is defined by the equation \(cx + dy = -6\). By selecting two different points on the plane and substituting them in, we find the specific coefficients (\(c\) and \(d\)) needed.

Some essentials about coordinate geometry:

• Coordinates \((x, y)\) represent a specific location on the plane.
• A straight line can be expressed as a linear equation, often in the form \(Ax + By = C\).
• Understanding the relationship between points and equations helps in interpreting and solving geometric problems.

This example demonstrates how points on a graph can be used to derive or confirm the equation of a line.

Solving Equations

Solving equations involves finding the values of variables that satisfy the equation. In linear equations, this often means manipulating and simplifying expressions to isolate the variable.

In our task, solving equations means:

• Substituting the values from the given points into the linear equation.
• Using algebraic methods to simplify the equations, such as adding or subtracting equations to eliminate variables.
• Simplifying further to find the values of unknowns, \(c\) and \(d\).

This problem required finding values for \(c\) and \(d\) by solving a simple two-variable system of equations which are often solved by methods like substitution or elimination.
Effective solving requires understanding equation properties, identifying symmetries and applying logical operations to reach a solution.

Points on a Line

Points on a line refer to the concept where given coordinates verify a line equation. If substituting the \((x, y)\) coordinates from a point into a line equation yields a true statement, it confirms the point lies on the line.

In this exercise, we check if the points \((1, 3)\) and \((-2, 12)\) lie on the same line by substituting them into the equation \(cx + dy = -6\) and solving for \(c\) and \(d\).

This relationship between points and lines is fundamental:

• A line in two-dimensional space can be uniquely determined if two distinct points on it are known.
• Substituting point coordinates in the line equation to check consistency is a common method to validate the equation of a line.
• Each point satisfying the line equation indicates consistency within the geometric rule defined by that line equation.

Understanding how points relate to line equations is vital in geometry and algebra for confirming linear relationships.