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Chapter 9: Problem 7

Write a solution to each equation using parameters. a. \(2 x-y=3\) b. \(x-2 y+z=0\)

Short Answer

Expert verified
Equation (a): \(x = \frac{y+3}{2}\), \(y = 2x - 3\). Equation (b): \(x = 2y - z\), \(y = \frac{x-z}{2}\), \(z = x - 2y\).

Step by step solution

01

Solving for x in Equation (a)

Take the equation from part (a), which is \(2x - y = 3\). Solve for \(x\) in terms of the parameter \(y\). Add \(y\) to both sides: \(2x = y + 3\). Then divide both sides by 2: \(x = \frac{y+3}{2}\).
02

Solving for y in Equation (a)

Take the same equation from part (a), \(2x - y = 3\). Solve for \(y\) in terms of the parameter \(x\). Subtract \(2x\) from both sides: \(-y = -2x + 3\). Multiply both sides by -1 to solve for \(y\): \(y = 2x - 3\).
03

Solving for x in Equation (b)

Take the equation from part (b), which is \(x - 2y + z = 0\). Solve for \(x\) in terms of the parameters \(y\) and \(z\). Add \(2y\) and subtract \(z\) from both sides: \(x = 2y - z\).
04

Solving for y in Equation (b)

Take the equation from part (b), \(x - 2y + z = 0\). Solve for \(y\) in terms of the parameters \(x\) and \(z\). Add \(z\) to both sides and divide the entire equation by 2: \(-2y = -x + z\). Multiply by -1 to get \(2y = x - z\), then divide by 2: \(y = \frac{x - z}{2}\).
05

Solving for z in Equation (b)

Take the equation from part (b), \(x - 2y + z = 0\). Solve for \(z\) in terms of \(x\) and \(y\). Subtract \(x\) and add \(2y\) to both sides to isolate \(z\): \(z = x - 2y\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Equations with Parameters
Solving equations with parameters involves expressing one variable as a function of other variables. Parameters are variables that define a family of solutions. For example, in the equation \(2x - y = 3\), we can express \(x\) or \(y\) as functions of each other. This allows a deeper understanding of how \(x\) and \(y\) relate and change with different values.To solve for \(x\) regarding \(y\), rearrange to get \(x = \frac{y+3}{2}\). Here, \(x\) is dependent on whatever value \(y\) takes. Similarly, solving for \(y\), we find \(y = 2x - 3\). This confirms how values alternate depending on the parameter chosen, providing flexibility in analyzing systems.
Two-variable Equations
Two-variable equations, like \(2x - y = 3\), are common in algebra. These linear equations show a relationship between two variables, \(x\) and \(y\), often forming a straight line when graphed. Solving these involves manipulating terms to isolate one variable鈥攕olving for either \(x\) or \(y\). The equation \(2x - y = 3\) shows how each variable interacts. You can express \(x\) as \(x = \frac{y + 3}{2}\) or \(y\) as \(y = 2x - 3\). This dual expression lets you calculate any value needed while keeping the other variable as a parameter.Key steps involve basic algebraic operations:
  • Addition and subtraction to isolate terms.
  • Division or multiplication to solve for the specific variable.
Three-variable Equations
Equations with three variables, such as \(x - 2y + z = 0\), add complexity but follow similar algebraic manipulation. These equations describe planes in three-dimensional space, and solving for a variable gives us family of lines or points as solutions.You can solve for any variable by expressing it in terms of the others. For example:
  • To find \(x\), rearrange to get \(x = 2y - z\).
  • For \(y\), rearrange to give \(y = \frac{x - z}{2}\).
  • To solve for \(z\), use \(z = x - 2y\).
These equations demonstrate how variables intertwine, affecting each other's values directly. They are useful in fields like physics and engineering, where multiple factors interact.

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Most popular questions from this chapter

a. Create a system of three equations in three unknowns that has \(x=-3\) \(y=4,\) and \(z=-8\) as its solution. b. Solve this system of equations using elementary operations.

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For the planes \(\pi_{1}: 3 x+4 y-12 z-26=0\) and \(\pi_{2}: 3 x+4 y-12 z+39=0,\) determine a. the distance between \(\pi_{1}\) and \(\pi_{2}\) b. an equation for a plane midway between \(\pi_{1}\) and \(\pi_{2}\) c. the coordinates of a point that is equidistant from \(\pi_{1}\) and \(\pi_{2}\)
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