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Chapter 4: Problem 54

Solve each equation for \(y\). $$ -2 x+y=9 $$

Short Answer

Expert verified
y = 2x + 9

Step by step solution

01

Isolate y

Start by isolating the term containing y. Move the term \( -2x \) to the other side of the equation by adding \( 2x \) to both sides.
02

Simplify the equation

After adding \( 2x \) to both sides, the equation becomes: \[ y = 2x + 9 \]

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Key Concepts

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

Isolate Variable
When solving linear equations, isolating the variable is a crucial step. This means you need to get the variable you are solving for, such as 饾懄, on one side of the equation by itself. In the given exercise, the initial equation is \[ -2x + y = 9 \]. To isolate 饾懄, we need to remove \[ -2x \] from the left side. We can do this by adding \[ 2x \] to both sides of the equation. This process changes the left side to \[ y \] and the right side to \[ 2x + 9 \]. So, after isolating 饾懄, your equation should look like this: \[ y = 2x + 9 \]. By isolating the variable, we make it easier to understand what values 饾懄 may take for different values of 饾懃.
Simplification
Simplification is the process of making an equation easier to understand and solve. After isolating the variable, we often need to simplify the equation further. In our example, after we isolate 饾懄 by adding \[ 2x \] to both sides, we get \[ y = 2x + 9 \]. Even though this equation is already quite simple, it is an essential part of solving linear equations.

Simplification can involve several steps such as:
  • Combining like terms
  • Reducing fractions
  • Eliminating complex fractions
  • Factoring expressions

These steps help to make sure the equation is in its simplest form, making it easier to understand and solve.
Linear Equations
Linear equations are equations of the first degree, meaning they involve only the first powers of the variables. They form straight lines when graphed on a coordinate plane. The general form of a linear equation in two variables, 饾懃 and 饾懄, is \[ ax + by = c \], where 饾憥, 饾憦, and 饾憪 are constants.

In this exercise, our linear equation is \[ -2x + y = 9 \]. This is already a form of a linear equation because it only involves 饾懃 and 饾懄 to the first power. Once we isolate 饾懄 and simplify, our final form \[ y = 2x + 9 \] remains a linear equation.
  • Graphing this equation would show a straight line.
  • The slope of the line is determined by the coefficient of 饾懃, which is 2 in this case.
  • The y-intercept, where the line crosses the y-axis, is given by the constant term, which is 9.

Understanding the properties of linear equations is key to solving them efficiently.

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Most popular questions from this chapter

Solve each equation for \(y\). $$ 5 x-3 y=12 $$

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