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Chapter 1: Problem 36

\(y=1+x\) \(2 x+y=-2\)

Short Answer

Expert verified
The solutions for the system of equations are \(x = -1\) and \(y = 0\).

Step by step solution

01

Write Down the Given Equations

The system of equations is \(y = 1 + x\) and \(2x + y = -2\).
02

Substitute y in the Second Equation

Substitute y from the first equation into the second equation. This gives us \(2x + (1 + x) = -2\). Simplifying this equation gives \(3x + 1 = -2\).
03

Solve for x

By subtracting 1 from both sides and then dividing by 3, this gives \(x = -1\).
04

Solve for y

In the first equation, substitute x = -1, it gives \(y = 1 + -1\), hence the solution is \(y = 0\).

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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 powerful technique for solving systems of equations. It involves replacing one variable in an equation with an expression from another equation. This method simplifies the problem by turning it into a single-variable equation. Here's how it works:
  • Solve one of the equations for one variable in terms of the other variables.
  • Substitute this expression into the other equation.
  • Solve the resulting equation for the single variable.
  • Finally, substitute the solution back into one of the original equations to find the remaining variable.
In our example:
  • We solved the first equation, turning it into an expression for y: \(y = 1 + x\).
  • This expression was substituted into the second equation, simplifying the problem to one variable, which was then easily solved.
This approach is ideal for linear equations because of their simplicity, but it can also be adapted for more complex systems.
Linear Equations
Linear equations form a straight line when graphed, hence the name "linear." They are one of the most foundational concepts in algebra, typically appearing in the form \((ax + by = c)\). Here, each variable is only raised to the first power, and the graph does not include curves or parabolas.Key characteristics of linear equations:
  • The graph of a linear equation is a straight line.
  • They have consistent rates of change, represented by the slope in the equation's graph.
  • These equations can have one or more variables, and each term is either a constant or a product of a constant and a variable.
In our system, both equations \(y = 1 + x\) and \(2x + y = -2\) are linear. They can be plotted on a coordinate plane and will intersect at the point \(x = -1, y = 0\). This point is the solution to the system, representing the values that satisfy both equations simultaneously.
Algebraic Manipulation
Algebraic manipulation is key to solving equations, especially when using techniques like substitution. It involves rearranging and simplifying equations to isolate variables and solve for unknowns.This encompasses a variety of techniques:
  • Combining like terms to simplify expressions, as seen when we combined \(2x + (1 + x)\) to \(3x + 1\).
  • Performing operations on both sides of the equation, like subtracting \(1\) to isolate terms.
  • Using properties of equality to maintain balance, such as the division that leads to the solution \(x = -1\).
In the original problem, we adeptly used these manipulations to simplify and solve the equations step-by-step. Mastering these skills allows for efficient problem-solving and makes tackling complex equations more approachable.

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Most popular questions from this chapter

Solve the system of linear equations using the substitution method. \(2 x-3 y+z=10\) \(y+2 z=13\) \(z=5\)

Solve the system using the elimination method. \(2 x+y-z=9\) \(-x+6 y+2 z=-17\) \(5 x+7 y+z=4\)

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