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Chapter 9: Problem 3

Determine whether \(x=-7, y=5,\) and \(z=\frac{3}{4}\) is a solution to the following systems: a. ? \(x-3 y+4 z=-19\) ? \(x-8 z=-13\) ? \(x+2 y=3\) b. ? \(3 x-2 y+16 z=-19\) ? \(3 x-2 y=-23\) ? \(8 x-y+4 z=-58\)

Short Answer

Expert verified
Point a is valid; point b is not a solution (second equation fails).

Step by step solution

01

Substitute into the First Equation of System a

Substitute the given values \(x = -7\), \(y = 5\), and \(z = \frac{3}{4}\) into the first equation of system a: \(x - 3y + 4z = -19\). Do the math: \(-7 - 3(5) + 4(\frac{3}{4}) = -7 - 15 + 3 = -19\). The equation holds true.
02

Check the Second Equation of System a

Now, substitute the same values into the second equation of system a: \(x - 8z = -13\). Calculate: \(-7 - 8(\frac{3}{4}) = -7 - 6 = -13\). This equation also holds true.
03

Verify the Third Equation of System a

Finally, substitute the values into the third equation: \(x + 2y = 3\). Check the result: \(-7 + 2(5) = -7 + 10 = 3\). This equation is correct as well.
04

Substitute into the First Equation of System b

Substitute the values into the first equation of system b: \(3x - 2y + 16z = -19\). Calculate: \(3(-7) - 2(5) + 16(\frac{3}{4}) = -21 - 10 + 12 = -19\). This equation is valid.
05

Verify the Second Equation of System b

Next, substitute into the second equation: \(3x - 2y = -23\). Calculate: \(3(-7) - 2(5) = -21 - 10 = -31\). This equation does not hold true as it should be \(-23\).
06

Check the Third Equation of System b

Finally, substitute into the third equation: \(8x - y + 4z = -58\). Check: \(8(-7) - 5 + 4(\frac{3}{4}) = -56 - 5 + 3 = -58\). Here, the equation holds true.

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Key Concepts

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

Solution Verification
When working with systems of equations, one essential skill is verifying whether a proposed solution is correct. This process ensures that the values given for the variables satisfy each equation in the system. You'll typically follow these steps to verify a solution:
  • Substitute the given values into each equation independently.
  • Simplify the equation to confirm whether both sides are equal.
  • If each equation in the system is satisfied, then the solution is correct.
  • If even one equation is not satisfied, the values do not make up a solution to the system.
In the given system, we had to check if the set of values \((x = -7, y = 5, z = \frac{3}{4})\) solves both systems of equations presented. We went equation by equation to confirm whether both sides matched, thereby determining the validity of the solution.
Substitution Method
The substitution method is one way to solve systems of equations. This method is helpful because it focuses on solving one equation for a single variable and then using that expression to substitute into the other equations. While our exercise focused on verification, understanding substitution helps in finding solutions when not given a potential answer initially. To use the substitution method:
  • Choose one equation and solve it for one variable in terms of the others.
  • Substitute this expression into the other equations.
  • Solve the resulting equations, which now have fewer variables.
  • Continue simplifying until you can find specific values for all variables.
The method is particularly useful when one equation is simple enough to express one variable directly in terms of the others. It helps reduce complexity and allows systematic elimination of variables.
Linear Equations
Linear equations are fundamental elements of algebra and form the backbone of systems of equations. A linear equation is of the form:\[ ax + by + cz = d \]where \(a\), \(b\), and \(c\) are coefficients and \(d\) is a constant term. Each equation describes a straight line in a two-dimensional space, and a plane in three-dimensional space, or more complex geometric objects in higher dimensions.When placed in a system:
  • Multiple linear equations interact and the solution set is where their graphs intersect.
  • Two linear equations intersect at a single point, multiple times depending on the dimension, or not at all.
  • Solving these systems often means finding the common intersection of all involved equations.
Understanding linear equations is crucial as they are applicable not only in mathematical contexts, but in many practical real-world scenarios. Their simplicity makes them a great starting point before progressing to more complex non-linear systems.

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Most popular questions from this chapter

Determine the value(s) of the constant \(k\) for which the following system of equations has a. no solutions b. one solution c. infinitely many solutions ? x+y=6 ? 2 x+2 y=k

Determine the following distances: a. the distance from \(A(3,1,0)\) to the plane with equation \(20 x-4 y+5 z+7=0\) b. the distance from \(B(0,-1,0)\) to the plane with equation \(2 x+y+2 z-8=0\) c. the distance from \(C(5,1,4)\) to the plane with equation \(3 x-4 y-1=0\) d. the distance from \(D(1,0,0)\) to the plane with equation \(5 x-12 y=0\) e. the distance from \(E(-1,0,1)\) to the plane with equation \(18 x-9 y+18 z-11=0\)

Explain why there is no solution to the following system of equations: ? \(2 x+3 y-4 z=-5\) ? \(x-y+3 z=-201\) ? \(5 x-5 y+15 z=-1004\)

Determine the solution to the following system of equations: ? \(\frac{1}{a}+\frac{1}{b}-\frac{1}{c}=0\) ? \(\frac{2}{a}+\frac{3}{b}+\frac{2}{c}=\frac{13}{6}\) ? \(\frac{4}{a}-\frac{2}{b}+\frac{3}{c}=\frac{5}{2}\)

Determine the distance between the following parallel lines: a. \(2 x-y+1=0,2 x-y+6=0\) b. \(7 x-24 y+168=0,7 x-24 y-336=0\)
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