Problem 79 - Algebra A Combined Function

Question

Answer true or false. Describe how to check to see if the graph of \(2 x-4 y=7\) passes through the points (1.4,-1.05) and (0,-1.75) . Then follow your directions and check these points.

Step-by-Step Solution

Answer

True; both points are on the line.

Step 1: Understanding the Equation

The given equation is a linear equation in the form of \[2x - 4y = 7\] This represents a straight line in the coordinate plane. To check if a point lies on this line, we substitute the x and y values of the point into the equation and see if the equation holds true.

Step 2: Substitute Point (1.4, -1.05) into the Equation

Substitute \(x = 1.4\) and \(y = -1.05\) into the equation:\[2(1.4) - 4(-1.05) = 7\]Calculate:\[2.8 + 4.2 = 7\]\[7 = 7\]The equation holds true, so the point (1.4, -1.05) lies on the line.

Step 3: Substitute Point (0, -1.75) into the Equation

Substitute \(x = 0\) and \(y = -1.75\) into the equation:\[2(0) - 4(-1.75) = 7\]Calculate:\[0 + 7 = 7\]\[7 = 7\]The equation holds true, so the point (0, -1.75) lies on the line.

Key Concepts

Coordinate Plane

A coordinate plane is a two-dimensional surface on which we can plot points, lines, and curves. It consists of two number lines that intersect at a right angle. The horizontal line is known as the x-axis, and the vertical line is known as the y-axis. These axes divide the plane into four quadrants.

The intersection of the x-axis and y-axis is called the origin, which has the coordinates (0,0). Each point on the plane is defined by a pair of numbers known as coordinates, written as (x,y), where x represents the horizontal position and y represents the vertical position.

Understanding the coordinate plane is crucial for graphing linear equations and determining if specific points lie on a line. In our given exercise, we are verifying if the points (1.4, -1.05) and (0, -1.75) are on the graph of the line represented by the equation. The coordinate plane provides the visual aid to view such graphs and points.

Substitution Method

The substitution method is a straightforward technique for solving equations, particularly useful for checking if certain points lie on the line represented by a linear equation.

To use this method, follow these steps:

• Take the coordinates of the point in question, which is given in the form (x, y).
• Substitute these x and y values into the linear equation.
• Simplify to see if the actual equation holds true (results in an identity like 7 = 7 in our example).

By substituting the values, you're essentially checking if the equation balances. If it does, the point indeed lies on the line. For example, after substituting (1.4, -1.05) into the equation \[2x - 4y = 7\], we find that it holds true. This means the point is on the line. Similarly, we verify point (0, -1.75). This method helps in understanding the consistency of the point with the equation's graph.

Points on a Graph

When we talk about points on a graph, we refer to specific locations on the coordinate plane that have definite x and y coordinates. These points can help illustrate the relationship described by an equation.

A linear equation, like \[2x - 4y = 7\], typically forms a straight line when graphed. If a point lies on this line, it means substituting the point's coordinates into the equation will result in a true statement.

• To determine the position of a point, take its coordinates (x, y).
• Plot these on the graph, using the x-axis for the x-value and the y-axis for the y-value.
• If the point is on the line, it will fall exactly on this graphed line.

In our exercise, both points, (1.4, -1.05) and (0, -1.75), lie on the graph of the equation. This confirmation comes from the fact that they satisfy the equation, validating their alignment on the graph. Points on a graph provide visual and analytical confirmation of the relationships expressed by the linear equation.