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Chapter 3: Problem 53

Find the slope of the line determined by each equation. $$2(y+1)=3 y$$

Short Answer

Expert verified
The slope of the line determined by the equation is 1.

Step by step solution

01

Simplify the equation

First start by distributing the 2 into the parentheses \(2(y+1)=3y\) which results in \(2y+2=3y\).
02

Solve for y

Next, move the term 2y to the other side of the equation to solve for y. The result \(2=3y-2y\) simplifies to \(2=y\).
03

Identify the slope of the line

Since the equation is in the form \(y=mx+c\) and the coefficient of y is 1, the slope of the line 'm' is 1.

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Key Concepts

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

Understanding Linear Equations
Linear equations are a fundamental aspect of algebra, representing relationships between variables that create straight lines when graphed. These equations are often written in the form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. This form is neat because it lets you instantly see the slope, which shows how steep the line is.

In linear equations, both the variables and their coefficients are simplified versions of real-life scenarios, like determining the rate of change of something.
  • The slope \(m\) is key in understanding how fast or slow a line rises.
  • The y-intercept \(c\) tells where the line crosses the y-axis.
Learning to extract and interpret these components makes graphing lines and solving problems much smoother.
Effective Techniques for Solving Equations
Solving equations means finding the values of variables that make the equation true. It's like unlocking a mystery using mathematical clues. For linear equations, your goal is usually to isolate \(y\) or \(x\) on one side of the equation.

There are a few strategies that you can use:

  • Distribute and Simplify: First, handle any parentheses by distributing numbers across terms inside them. This makes the equation easier to work with, just like we saw in the exercise when converting \(2(y+1)=3y\) to \(2y+2=3y\).
  • Combine Like Terms: Group similar terms together, especially those involving variables, which helps simplify the problem further.
  • Isolate the Variable: Move terms around to get the variable you want to solve by itself. Think of this as rearranging parts of a puzzle until you see the clear solution.
With practice, these techniques become intuitive and powerful tools.
Mastering Algebraic Manipulation
Algebraic manipulation involves skillfully rearranging and simplifying expressions to reveal the values you need. It's about seeing the equation from different angles until you see a clearer path to the solution. This is crucial when tackling more complex equations.

Here are some tips:
  • Maintain Balance: Whatever operation you do to one side of an equation, you do to the other. This keeps the equation balanced and valid.
  • Rearrange Thoughtfully: Shift terms across the equation to get closer to your goal. In our example, shifting \(2y\) helped simplify \(2y+2=3y\) to \(2=y\).
  • Break it Down: Simplify complex expressions step by step rather than all at once. Smaller steps can lead to fewer mistakes and clearer thinking.
Understanding and mastering these manipulations make complex problems much more approachable and can build a strong foundation for advanced math topics.

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Most popular questions from this chapter

Express each sentence as a formula. The current \(I\) varies directly with the voltage \(V\) and inversely with the resistance \(R\).

Fill in the blanks. Assume that \(k\) is a constant. The equation \(y=k x z\) represents _____ variation.

If points \(P(a, b)\) and \(Q(c, d)\) are two points on a rectangular coordinate system and point \(M\) is midway between them, then point \(M\) is called the midpoint of the line segment joining \(P\) and \(Q .\) (See the illustration on the following page. To find the coordinates of the midpoint \(M\left(x_{M}, y_{M}\right)\) of the segment PQ, we find the average of the \(x\) -coordinates and the average of the \(y\)-coordinates of \(P\) and \(Q\). $$x_{M}=\frac{a+c}{2}$$ and $$y_{M}=\frac{b+d}{2}$$ Find the coordinates of the midpoint of the line segment with the given endpoints. $$A(-2, y+1) \text { and } B(6, y-1)$$

Graph each equation. SEE EXAMPLE 6. (OBJECTIVE 5 ) $$x=5$$

Give examples of an equation in one variable and an equation in two variables. How do their solutions differ?
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