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Chapter 13: Problem 14

Solve by substitution. \(\begin{aligned} y &=2-3 x \\ 6 x+2 y &=7 \end{aligned}\)

Short Answer

Expert verified
The solution to the system of equations is \( x = \frac{3}{2}, y = -\frac{1}{2} \)

Step by step solution

01

Substitute y into second equation

Substitute \( y = 2 - 3x \) into \( 6x + 2y = 7 \), getting: \( 6x + 2(2 - 3x) = 7 \)
02

Simplify the equation to find x's value

Simplify the equation to \( 6x + 4 - 6x = 7 \), then isolate x, giving us \( x = \frac{3}{2} \)
03

Substitute x into initial equation to find y's value

Substitute \( x = \frac{3}{2} \) into \( y = 2 - 3x \), giving us \( y = -\frac{1}{2} \)

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Key Concepts

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

Understanding the Substitution Method
The substitution method is a powerful technique for solving systems of equations. It works by expressing one variable in terms of another and then substituting this expression into another equation. This changes the system of equations into a manageable single-variable equation.
In the given exercise, we start with two equations:
  • \( y = 2 - 3x \)
  • \( 6x + 2y = 7 \)
To apply the substitution method:
  • First, solve one of the equations for one variable, as has been done with \( y = 2 - 3x \).
  • Then, substitute this expression into the other equation, effectively removing one variable from the equation.
This is the heart of the substitution method: substituting to simplify. Through substitution, a system of two variables is reduced to one, making it much easier to solve systematically.
Exploring Linear Equations
Linear equations are equations of the first degree. This means each term is either a constant or the product of a constant and a single variable. Linear equations graph as straight lines on the coordinate plane.
For the system given in the exercise:
  • The first equation, \( y = 2 - 3x \), is already in slope-intercept form. This makes it easy to visualize its graph.
  • The second equation can be rearranged or used directly depending on the method of solving (such as substitution).
Linear systems, like the one in this problem, represent lines on a graph. The point where these lines intersect is the solution to the system. This point, found via substitution, represents the values of \( x \) and \( y \) that satisfy both equations simultaneously.
Understanding Algebraic Manipulation
Algebraic manipulation involves the use of various techniques to simplify, rearrange, or solve equations. It's a fundamental skill in solving equations and understanding mathematical relationships.
In the provided problem, after substituting \( y = 2 - 3x \) into the second equation, we carry out several algebraic manipulations:
  • Begin with the substitution: \( 6x + 2(2 - 3x) = 7 \).
  • Distribute the 2 into the expression: \( 6x + 4 - 6x = 7 \).
  • Simplify the equation by combining like terms: in this case, the \( 6x \) terms cancel out.
  • Isolate the variable by solving \( 4 = 7 \) to find \( x = \frac{3}{2} \).
Once \( x \) is found, substitute back into an earlier equation to solve for \( y \). These steps demonstrate the power of algebraic manipulation in simplifying and solving equations.

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Most popular questions from this chapter

Solve by using a graphing calculator. $$\begin{aligned} &x=2 y-5\\\ &x=3 y+2 \end{aligned}$$

Rewrite equation so that the coefficients are integers. Then solve the system of equations by the substitution method. \(1.25 x-0.01 y=1.5\) \(0.24 x-0.02 y=-1.52\)

Solve by graphing. $$\begin{aligned} &x=3\\\ &y=-2 \end{aligned}$$ (THE GRAPH CANNOT COPY)

Solve by the addition method. $$\begin{aligned} &2 x+3 y=7-2 x\\\ &7 x+2 y=9 \end{aligned}$$

Solve by substitution. \(\begin{aligned} y &=3 \\ 3 x-2 y &=6 \end{aligned}\)
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