Problem 45 - Algebra Form and Function

Question

Match the statements with equations \(\mathrm{I}-\) VI. III. \(y=5 x+30\) IV. \(\quad y=-5(6-x)\) V. \(y=\frac{2 x+90}{3}\) VI. \(\quad y=-\frac{2}{3}(x-8)+20\). These two lines have the same \(y\) -intercept.

Step-by-Step Solution

Answer

Answer: Equations III and V have the same y-intercept.

Step 1: Find the y-intercept of Equation III.

For Equation III, substitute x = 0 and find y: \(y = 5(0) + 30 = 0 + 30 = 30\). The y-intercept is 30.

Step 2: Find the y-intercept of Equation IV.

For Equation IV, substitute x = 0 and find y: \(y = -5(6 - 0) = -5(6) = -30\). The y-intercept is -30.

Step 3: Find the y-intercept of Equation V.

For Equation V, substitute x = 0 and find y: \(y = \frac{2(0) + 90}{3} = \frac{90}{3} = 30\). The y-intercept is 30.

Step 4: Find the y-intercept of Equation VI.

For Equation VI, substitute x = 0 and find y: \(y = -\frac{2}{3}(0 - 8) + 20 = -\frac{2}{3}(-8) + 20 = \frac{16}{3} + 20 = \frac{76}{3}\). The y-intercept is \(\frac{76}{3}\).

Step 5: Compare the y-intercepts and find the matching lines

Based on our calculations: - Equation III: y-intercept = 30 - Equation IV: y-intercept = -30 - Equation V: y-intercept = 30 - Equation VI: y-intercept = \(\frac{76}{3}\) We can see that Equations III and V have the same y-intercept of 30. Therefore, these are the two lines that have the same y-intercept.

Key Concepts

Linear Equations

Linear equations are equations of the first degree, meaning they have variables raised only to the power of one. A standard form of a linear equation with two variables, say \(x\) and \(y\), is given by:

• \(ax + by + c = 0\)
• Or the slope-intercept form: \(y = mx + b\)

The slope, \(m\), indicates the line's steepness, while the y-intercept, \(b\), represents the point where the line crosses the y-axis. Linear equations graph as straight lines on a coordinate plane.
Considering the exercise above, we are using equations in the slope-intercept form to find specific values by substituting \(x\) or \(y\). This substitution helps us identify specific points on the line or solve for one variable when the other is known.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, uses a coordinate system to study geometric objects. The main tool here is the Cartesian coordinate system which has an \(x\)-axis (horizontal) and a \(y\)-axis (vertical).
In solutions like the one given, the Cartesian plane helps locate points like the y-intercept, where the line intersects the y-axis. To find a y-intercept from an equation, substitute \(x = 0\) and solve for \(y\).

• The y-intercept of \(y = 5x + 30\) is 30, meaning the line cuts the y-axis at the point (0, 30).
• Similarly, for \(y = \frac{2x + 90}{3}\), substituting \(x = 0\) gives a y-intercept of 30.

Coordinate geometry provides a visual and analytical way to understand relationships between equations and their graphical representations.

Comparison of Equations

When comparing linear equations, one key aspect is their y-intercepts. These points tell us where the lines intersect the y-axis and help determine if two lines are parallel, identical, or intersecting.
To identify lines with the same y-intercept, simply calculate the y-intercept for each equation by setting \(x = 0\), as shown in the solution steps. Equations that yield the same y-intercepts intersect the y-axis at the same point.
For example, in the original exercise, both Equations III \((y = 5x + 30)\) and V \((y = \frac{2x + 90}{3})\) share a y-intercept of 30.

• This means if these lines were plotted on the same graph, they would both cross the y-axis at \(y = 30\).

Comparing equations involves analyzing and contrasting these characteristics to understand how the lines relate to one another.