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Chapter 4: Problem 16

Solve each system of equations by the substitution method. See Examples I through 6 $$ \left\\{\begin{array}{l} {3 y-x=6} \\ {4 x+12 y=0} \end{array}\right. $$

Short Answer

Expert verified
The solution is \( x = -3 \) and \( y = 1 \).

Step by step solution

01

Choose an Equation for Substitution

We start by isolating one variable in the first equation, which is simpler. The first equation is \( 3y - x = 6 \). Let's solve for \( x \). Add \( x \) to both sides to get \( 3y = x + 6 \). Then subtract 6 from both sides: \( x = 3y - 6 \).
02

Substitute the Expression into the Second Equation

Now substitute the expression for \( x \) from Step 1 into the second equation, \( 4x + 12y = 0 \). Replace \( x \) with \( 3y - 6 \): \( 4(3y - 6) + 12y = 0 \).
03

Simplify the Substituted Equation

Distribute 4 in the substituted equation: \( 12y - 24 + 12y = 0 \). Combine like terms: \( 24y - 24 = 0 \).
04

Solve for y

Add 24 to both sides: \( 24y = 24 \). Then divide both sides by 24 to isolate \( y \): \( y = 1 \).
05

Substitute Back to Find x

Substitute \( y = 1 \) back into the expression for \( x \) from Step 1: \( x = 3(1) - 6 \). Simplify this: \( x = 3 - 6 = -3 \).
06

Write the Solution

The solution to the system of equations is \( x = -3 \) and \( y = 1 \). This means the point of intersection of the lines represented by the equations is \((-3, 1)\).

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Key Concepts

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

System of Equations
A system of equations is a collection of two or more equations with a common set of variables. In our case, the system is composed of two linear equations:
  • \( 3y - x = 6 \)
  • \( 4x + 12y = 0 \)
These equations are interrelated because they must be solved together, meaning we need to find values for the variables that satisfy both equations at the same time. The solution to this system is the point where the two lines, represented by these equations, intersect on a graph. Understanding how a system of equations works can help us see the robust relationship between variables, offering a visual insight into the scenarios they represent.
Linear Equations
Linear equations form straight lines when graphed and have variables raised only to the first power, their standard form being \( ax + by = c \). The equations provided in our system are both linear. Let's break them down:
  • The first equation, \( 3y - x = 6 \), can be rearranged to highlight its linear nature as \( x = 3y - 6 \). This shows \( x \) linearly dependent on \( y \).
  • The second equation, \( 4x + 12y = 0 \), can also be formatted in a slope-intercept form to easily graph it: \( x = -3y \).
Linear equations are critical in illustrating constant relationships between variables, showcasing how variables change in a predictable fashion. The intersection point of these lines represents the solution to the system.
Problem Solving
Solving systems of equations requires strategic approaches to find common variable values. The substitution method is one such strategy. Here's why it is effective:
  • It allows us to express one variable in terms of another, simplifying the problem. For example, from \( 3y - x = 6 \), we derived \( x = 3y - 6 \).
  • This substitution is then used in the other equation, reducing the number of variables to one, making the math simpler by solving a single variable equation.
  • Step by step, this approach narrows down our options, eventually leading us to a unique solution that satisfies both original equations.
Understanding these procedural steps enhances our problem-solving toolkit, enabling us to tackle more complex systems with confidence.
Solution Steps
Breaking down the solution into clear steps makes solving systems manageable. Here's a recap of the substitution method applied in our example:
  • Step 1: Choose an Equation for Substitution: Isolate one variable, say \( x = 3y - 6 \) from \( 3y - x = 6 \).
  • Step 2: Substitute: Insert \( x = 3y - 6 \) into the second equation to form \( 4(3y - 6) + 12y = 0 \).
  • Step 3: Simplify and Solve for \( y \): Combine and solve \( 24y - 24 = 0 \) yielding \( y = 1 \).
  • Step 4: Find \( x \): Use \( y \)'s value, substitute back to get \( x = -3 \).
  • Step 5: Conclude the Solution: The intersection is at \( (-3, 1) \), showing solutions \( x = -3 \) and \( y = 1 \).
This organized approach to solving the system ensures that each step builds logically upon the previous one, leading to the accurate solution of the equations.

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Most popular questions from this chapter

Solve each system of linear equations by graphing. See Examples 3 through 6 $$ \left\\{\begin{array}{l} {x=3} \\ {y=-1} \end{array}\right. $$

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