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Chapter 4: Problem 5

If \(2 x+3 y=20\) and \(6 x+2 y=46\) then \(y\) is: A. 5 B. 7 C. 2 D. 9 E. Cannot be determined

Short Answer

Expert verified
C. 2

Step by step solution

01

Write down the system of equations

The given system of equations is: 1) 2x + 3y = 20 2) 6x + 2y = 46
02

Simplify the equations

To simplify the second equation (2), divide each term by 2: \(\frac{6x + 2y}{2} = \frac{46}{2}\) This gives us: 3x + y = 23
03

Solve for y

From the simplified equation 3x + y = 23, isolate y: y = 23 - 3x
04

Substitute y into the first equation

Substitute the value of y into the first equation (2x + 3y = 20): 2x + 3(23 - 3x) = 20 Solve for x: 2x + 69 - 9x = 20 -7x = -49 x = 7
05

Plug the value of x back into y = 23 - 3x

Substitute x = 7 into y = 23 - 3x: y = 23 - 21 y = 2

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Key Concepts

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

Linear Equations
Linear equations are equations involving constants and variables where the variables are raised to the power of one. They generally take the form: ax + by = c, where a, b, and c are constants. For instance, in the exercise given, the linear equations are 2x + 3y = 20 and 6x + 2y = 46. Linear equations are straightforward since their solutions yield straight lines when graphed on a coordinate plane. Key characteristics include having a single solution, infinitely many solutions, or no solutions at all, based on whether their lines intersect, overlap, or are parallel, respectively.
System of Equations
A system of equations comprises two or more equations with the same set of variables. In our problem, we have two equations: 2x + 3y = 20 and 6x + 2y = 46. The main goal is to find values for the variables that satisfy all equations simultaneously. To solve such systems, you can use methods like substitution or elimination. In this case, we simplified the second equation by splitting it by 2, then used substitution to solve for y by isolating x in one of the equations and substituting into the other. This method helps to systematically eliminate variables and solve the system.
Algebra
Algebra involves manipulating variables and constants using arithmetic operations to solve equations or find unknowns. In the provided solution, we practiced simplifying equations and combining like terms. Key algebraic steps include:
  • Isolating variables: Simplifying the second equation to 3x + y = 23 allowed us to isolate y
  • Substitution: Substituting y back into the first equation helped eliminate terms and solve for x
  • Simplification: Breaking down equations like 2x + 69 - 9x = 20 to -7x = -49 simplifies the solution process
Overall, algebraic manipulation was crucial to step-by-step solving of the exercise. Such competence is fundamental for higher math and everyday problem-solving.

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