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Chapter 5: Problem 6

Solve each system by substitution. $$\begin{aligned}&x=y-8\\\&-3 x-y=12\end{aligned}$$

Short Answer

Expert verified
The solution to the system of linear equations is the point of intersection: \((-5, 3)\).

Step by step solution

01

Solve one equation for either x or y

The first equation is already solved for x: \(x = y - 8\)
02

Substitute the obtained value from Step 1 into the second equation

Substitute the value of x (\(y - 8\)) into the second equation: \(-3(x) - y = 12\) \(-3(y - 8) - y = 12\)
03

Solve the second equation for y

Now, we will solve the second equation after substitution for y: \(-3(y - 8) - y = 12\) \(-3y + 24 - y = 12\) \(-4y + 24 = 12\) Now, subtract 24 from both sides: \(-4y = -12\) Now, divide both sides by -4: \(y = 3\)
04

Substitute the obtained value of y back into the first equation and solve for x

We'll substitute the obtained value for y back into the first equation and solve for x: \(x = y - 8\) \(x = 3 - 8\) Now, subtract 3 from both sides: \(x = -5\)
05

Write the solution as an ordered pair (x, y)

The solution to the system of linear equations is: \((x, y) = (-5, 3)\) So, the system's solution is the point of intersection: \((-5, 3)\).

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Key Concepts

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

Substitution Method
The substitution method is a straightforward way to solve a system of linear equations. It involves isolating one variable in one equation and then substituting that expression into the other equation. This technique gradually transforms a system of two equations into a single equation with one variable, making it easier to solve.Let's break this down further:
  • Pick one of the equations and solve for one variable, either \(x\) or \(y\). For instance, from \(x = y - 8\), we already have \(x\) expressed in terms of \(y\).
  • Substitute this expression into the other equation. In our example, replace \(x\) in \(-3x - y = 12\) with \(y - 8\).
  • This substitution yields a new equation with only one variable to solve, simplifying the problem significantly.
The main advantage of the substitution method is its simplicity, especially for systems where one equation is already solved for a variable.
Linear Equations
Linear equations are mathematical statements that describe a straight line when graphed. They are fundamental concepts in algebra and appear in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables.Key features of linear equations include:
  • The highest exponent of the variables is one, giving it a linearity property.
  • Simpler than non-linear equations due to their constant rate of change.
  • They often represent real-world relationships such as distance over time or financial arrangements.
In solving systems like our example, understanding that each equation represents a line facilitates finding points where these lines intersect, which are their solution.
Solving for Variables
Solving for variables is the process of determining the values of unknowns in equations. In a system of linear equations, solving for variables involves finding a specific numerical solution (or set of solutions) that satisfy all the equations simultaneously.Here's a breakdown of how we generally solve for variables:
  • Use algebraic manipulation like adding, subtracting, multiplying, or dividing terms to isolate the variable.
  • In our exercise, we solved \(-3(y - 8) - y = 12\) for \(y\) by simplifying and rearranging terms to find \(y = 3\).
  • Once one variable is found, substitute it back into the other equation to find the second variable. Here, substituting \(y\) back gives us \(x = -5\).
  • The final solution represents the intersection point of lines described by the equations, providing a visual correspondence as well as a numerical solution.
Understanding how to methodically solve for variables is essential for mastering linear systems, making future problems more approachable and less intimidating.

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Most popular questions from this chapter

(4.6) Determine if each pair of lines is parallel, perpendicular or neither. $$\begin{aligned} &x+2 y=22\\\ &2 x-y=0 \end{aligned}$$

Write the slope-intercept form of the equation of the line, if possible, given the following information. $$m=\frac{1}{2} \text { and contains }(8,-3)$$

Solve each system \(\begin{aligned} 2 a-3 b &=-4 \\ 3 b-c &=8 \\\\-5 a+4 c &=-4 \end{aligned}\)

(4.7) Identify the domain and range of each relation, and determine whether each relation is a function. $$\\{(-4,-9),(0,3),(2,9),(5,18)\\}$$

Write an equation of the line parallel to the given line and containing the given point. Write the answer in slopeintercept form or in standard form, as indicated. \(y=-3 x+1 ;(5,-19) ;\) slope-intercept form
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