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Chapter 2: Problem 45

Find \(y\) given that \(x=3\). $$y=-2 x+5$$

Short Answer

Expert verified
y = -1

Step by step solution

01

- Substitute the value of x

Given the equation \( y = -2x + 5 \) and the value of \( x = 3 \), substitute 3 in place of \( x \).
02

- Simplify the expression

After substituting, the equation becomes \( y = -2(3) + 5 \). Now, simplify the expression inside the parenthesis first.
03

- Compute the product

Calculate \( -2 \times 3 \) to get \( -6 \).
04

- Add the constant term

Add the result from Step 3 to the constant term: \( -6 + 5 \). This simplifies to \( -1 \).
05

- Write down the final value of y

The value of \( y \) is therefore \( -1 \).

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Key Concepts

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

substitution in algebra
Substitution in algebra is a technique where you replace a variable with a given value. This is a handy method to solve equations when one variable's value is already known. To use substitution in the provided exercise, we replaced the variable \( x \) with the value \( 3 \). By doing this, we can transform the original equation, \( y = -2x + 5 \), into a simpler expression that can be solved easily. This helps in finding the value of \( y \) corresponding to \( x = 3 \).
For example:
  • Original Equation: \( y = -2x + 5 \)
  • Given \(x = 3 \)
  • New Equation (after substitution): \( y = -2(3) + 5 \)
In simpler terms, substitution allows us to convert and solve equations step by step.
simplifying expressions
Simplifying expressions is about making a mathematical expression as easy to work with as possible. This involves combining like terms and performing arithmetic operations. In the given problem, after substituting \(3\) for \( x \) in the equation \( y = -2x + 5 \), we obtained a new expression: \( y = -2(3) + 5 \).
To simplify:
  • First, compute the product within the parenthesis: \( -2 \times 3 = -6 \).
  • After this, add the constant term: \( -6 + 5 \).
  • This results in: \( -1 \).
Simplifying expressions like these help us understand the relationship between different parts of the equation.By simplifying each part step-by-step, it becomes easier to solve and understand the equation.
linear equations
Linear equations are algebraic equations in which each term is either a constant or the product of a constant and a single variable. These equations form straight lines when graphed on a coordinate plane. The given equation, \( y = -2x + 5 \), is a perfect example of a linear equation with \( y \) and \( x \) as its variables.
To solve a linear equation:
  • Identify the equation format: \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.
  • Substitute the given value of the variable (e.g., \( x = 3 \).
  • Simplify the resulting expression by performing basic arithmetic operations.
In the problem, we substituted \( x \) with \( 3 \) to find \( y \), simplified the expression, and ended up with \( y = -1 \). This method shows how straightforward linear equations can be when we understand the core concepts of substitution and simplification.

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Most popular questions from this chapter

Solve each inequality. Write the solution set using interval notation. $$1<|x+2|$$

Show a complete solution to each problem. Spring break. Fernell and Dabney shared the driving to Florida for spring break. Fernell averaged \(50 \mathrm{mph}\), and Dabney averaged 64 mph. If Fernell drove for 3 hours longer than Dabney but covered 18 miles less than Dabney, then for how many hours did Fernell drive?

Solve each absolute value inequality and graph the solution set. See Examples 5鈥�7. $$-2|5-x| \geq-14$$

If 3 is added to every number in \((4, \infty),\) the resulting set is \((7, \infty) .\) In each of the following cases, write the resulting set of numbers in interval notation. Explain your results. a) The number \(-6\) is subtracted from every number in \([2, \infty)\) b) Every number in \((-\infty,-3)\) is multiplied by 2 c) Every number in \((8, \infty)\) is divided by 4 d) Every number in \((6, \infty)\) is multiplied by \(-2\) e) Every number in \((-\infty,-10)\) is divided by \(-5\)

Identify the variable and write an inequality that describes situation. Burt is no taller than 5 feet.
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