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z=15x2+10xy6x4y For which of the ordered pairs, (x,y), below is z0? a. (-3,2) b. (-2,3) c. (25,0) d. (25,10)

Short Answer

Expert verified
The ordered pair (3,2) makes zeq0.

Step by step solution

01

- Understand the problem

Given a function of two variables, z=15x2+10xy6x4y, determine for which of the provided ordered pairs (x,y), the value of z is not equal to zero.
02

- Substitute pair (-3, 2)

Substitute x=3 and y=2 into the equation. z=15(3)2+10(3)(2)6(3)4(2) z=15(9)+10(6)+188 z=13560+188 z=85Since 85eq0, the pair (3,2) makes zeq0.
03

- Substitute pair (-2, 3)

Substitute x=2 and y=3 into the equation. z=15(2)2+10(2)(3)6(2)4(3) z=15(4)+10(6)+1212 z=6060+1212 z=0Since 0=0, the pair (2,3) does not make zeq0.
04

- Substitute pair (25,0)

Substitute x=25 and y=0 into the equation. z=15(25)2+10(25)(0)6(25)4(0) z=15(425)+0125+0 z=6025125 z=125125 z=0 Since 0=0, the pair (25,0) does not make zeq0.
05

- Substitute pair (25,10)

Substitute x=25 and y=10 into the equation. z=15(25)2+10(25)(10)6(25)4(10) z=15(425)+10(4)12540 z=6025+4012540 z=2.4+402.440 z=0 Since 0=0, the pair (25,10) does not make zeq0.

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Key Concepts

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

Variable Substitution
Variable substitution is a crucial technique in solving mathematical problems, especially in algebra. This concept involves replacing variables in an equation with given numerical values.
For example, in the given problem, we have the function of two variables: z=15x2+10xy6x4y. To check which ordered pair results in zeq0, we substitute the values of x and y from each pair into the equation:
  • For the pair (x,y)=(3,2), we substitute x=3 and y=2 into the equation.
  • For the pair (x,y)=(2,3), substitute x=2 and y=3.
  • For the pair (x,y)=(25,0), substitute x=25 and y=0.
  • For the pair (x,y)=(25,10), substitute x=25 and y=10.
With each substitution, you replace the variables in the function, then simplify to check if z results in a non-zero value.
Quadratic Equations
Quadratic equations are algebraic expressions where the highest power of the variable is squared. In this problem, the term 15x2 indicates that we are dealing with a quadratic equation in one of the variables.
When substituting values, you can notice how the squared term affects the overall value of z:
Formula example with substitution:
For (x,y)=(3,2), the term 15x2 becomes 15(3)2=135. This significant value has a large impact on z.

Due to the squared term, small changes in x can cause large changes in z, highlighting the importance of careful substitution and simplification when dealing with quadratic components of an equation.
Algebraic Manipulations
Algebraic manipulations refer to the processes of rearranging and simplifying expressions. In our problem, after substituting the values into the equation, we perform several steps to simplify the expressions:
  • Expand any squared terms, like (3)2=9.
  • Multiply constants and variables, such as 15(9) and 10(6).
  • Combine like terms, which means adding or subtracting terms with the same variables.
For example, in Step 2, the calculation process is:
15(3)2=135 10(3)(2)=60 6(3)=18 4(2)=8 Combiningthesegives:13560+188=85

This manipulation shows how each component of the equation contributes to the final value of z.
Problem-Solving Techniques
Effective problem-solving techniques involve understanding the problem, systematic substitution, careful simplification, and evaluation of results. Here's how you can approach this type of problem:
  • First, interpret the problem to understand what is required – in this case, finding when zeq0.
  • Next, substitute the given pairs (x,y) into the function one at a time.
  • Simplify the resulting equations through careful algebraic manipulations and combine like terms.
  • Finally, evaluate whether the simplified result is zero or not, confirming if the pair makes zeq0.
These steps not only help solve specific problems but also build a strong foundation in handling various algebraic challenges, fostering a deeper understanding of algebraic manipulations and quadratic equations.

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