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Chapter 12: Problem 29

Is \((3,4)\) a solution of \(y=-x+7 ?\)

Short Answer

Expert verified
Yes, the point (3,4) is a solution to the equation \(y=-x+7\).

Step by step solution

01

Understand the coordinates

The given point is (3,4). In a coordinate pair, the first value is the x-coordinate and the second value is the y-coordinate. Therefore, in this case, x=3 and y=4.
02

Substitute the values into the equation

Replace the values of x and y in the equation \(y=-x+7\). Thus the equation becomes \(4=-3+7\).
03

Solve the equation

Solve the equation. -3+7 equals 4, which is equal to y-value we have. Therefore, the point (3,4) does satisfy the equation \(y=-x+7\).

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Key Concepts

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

Linear Equations
Linear equations, such as the one you've encountered in the problem \(y = -x + 7\), represent a straight line when graphed on a coordinate plane. These equations have a general form \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept. The slope \(m\) shows how steep the line is, and the y-intercept \(b\) is the point where the line crosses the y-axis.
  • In our specific equation \(y = -x + 7\), the slope \(m\) is \(-1\), which means the line decreases one unit in \(y\) for every one unit increase in \(x\).
  • The y-intercept \(b\) is \(7\), so the line crosses the y-axis at the point \((0,7)\).
This concept helps in predicting the behavior of the line and determining which points lie on it. By understanding how to manipulate and interpret these variables, you can easily solve for unknown values or verify if a point, like \((3,4)\), lies on the line.
Coordinate Plane
The coordinate plane is a two-dimensional plane featuring a horizontal line (x-axis) and a vertical line (y-axis), intersecting at a point called the origin. Each point on the plane is defined by an ordered pair \((x, y)\), representing the horizontal and vertical distances from the origin.
  • The x-axis is the horizontal number line where values increase as you move right and decrease as you move left.
  • The y-axis is the vertical number line where values increase as you move up and decrease as you move down.
  • The origin, \((0,0)\), is where the x-axis and y-axis meet.
In the problem, the point \((3, 4)\) denotes a position three units to the right of the origin and four units up. This concept is crucial for visualizing equations graphically and understanding the spatial relationships between points.
Graphing Points
Graphing points on a coordinate plane involves plotting points based on their \((x, y)\) values. Each point is located by its distance from the origin, specified by the x-coordinate (horizontal) and y-coordinate (vertical).
  • To graph a point like \((3, 4)\), start at the origin \((0, 0)\).
  • Move horizontally to the right by 3 units to reach x=3.
  • From there, move vertically upward by 4 units to reach y=4.
Mark this position on the plane. This exercise is valuable to understand which points satisfy a given equation by observing where they lie relative to the graph of the equation. In this exercise, since the point \((3, 4)\) lies on the line represented by the equation \(y = -x + 7\), it is indeed a solution to the equation.

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Most popular questions from this chapter

Determine whether the line through \(P_{1}\) and \(P_{2}\) is parallel, perpendicular, or neither parallel nor perpendicular to the line through \(Q_{1}\) and \(Q_{2}\). $$P_{1}(1,-1), P_{2}(3,-2) ; Q_{1}(-4,1), Q_{2}(2,-5)$$

Determine whether the line through \(P_{1}\) and \(P_{2}\) is parallel, perpendicular, or neither parallel nor perpendicular to the line through \(Q_{1}\) and \(Q_{2}\). $$P_{1}(7,-1), P_{2}(-4,6) ; Q_{1}(3,0), Q_{2}(-5,3)$$

Graph by using the slope and \(y\) -intercept. (GRAPH CAN'T COPY) $$y=\frac{2}{3} x$$

Use the function \(f(x)=x^{2}-4 .\) For the given condition on \(a\) determine whether \(f(a)\) must be positive, must be negative, or could be either positive or negative. $$a<-2$$

Find the slope and \(y\) -intercept of the graph of the equation. $$2 x+3 y=8$$
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