Problem 26 - Algebra 1

Question

Find the \(y\) -intercept of the graph of the equation. $$ y=-2 x+5 $$

Step-by-Step Solution

Answer

The y-intercept of the equation \(y = -2x + 5\) is 5.

Step 1: Understand the y-intercept

The \(y\)-intercept of a graph is the point where the line crosses the \(y\)-axis, which occurs when \(x = 0\).

Step 2: Substitute x = 0

\(y = -2(0) + 5 = 5\)

Step 3: State the answer

The \(y\)-intercept of the graph of \(y = -2x + 5\) is \(5\). The line crosses the \(y\)-axis at the point \((0, 5)\).

Key Concepts

Understanding Linear Equations

Linear equations are mathematical expressions that form straight lines when graphed on a coordinate plane. A linear equation can typically be identified by the highest degree of variables being one, making it a polynomial of degree one. In a basic linear equation like the one presented, which is in the form of \( y = mx + c \), the relationship is straightforward. These equations model a constant rate of change or a simple proportional relationship between two variables, often represented as \(x\) and \(y\). Examples of linear equations include:

• \( y = 2x + 3 \)
• \( 4x + 5y = 20 \)
• \( y = -3x \)

New LineLinear equations are foundational in algebra and are used to describe various real-world relationships, such as distance over time or cost associated with quantity. Understanding the structure of these equations is crucial to solving them effectively.

Exploring Slope-Intercept Form

The slope-intercept form of a linear equation is a way of writing the equation such that it is easiest to identify the slope and the y-intercept, essential components for graphing. The form is:\[ y = mx + b \]where:

• \(m\) represents the slope, which describes the rate of change and the slant of the line.
• \(b\) represents the y-intercept, which is the point where the line crosses the y-axis.

The slope-intercept form simplifies the process of drawing a line and analyzing its characteristics. Knowing the y-intercept off the bat, like in the equation \(y = -2x + 5\), means you can immediately plot the starting point on the y-axis (in this case, \(b = 5\)). The slope, \(m = -2\), indicates that for every unit increase in \(x\), \(y\) decreases by 2 units.

Graphing Equations Made Simple

Graphing equations, particularly linear ones, can be an intuitive and enlightening process. When you have an equation in slope-intercept form, graphing becomes a step-by-step task:Start with the y-intercept. In the equation \(y = -2x + 5\), the y-intercept is 5. Place a point on the y-axis at \((0, 5)\).
After plotting the y-intercept, use the slope to determine the next points. Since the slope \(m = -2\) means you move 2 units downwards for every 1 unit you move right, from \((0, 5)\) you move 1 right to \((1, 3)\). Connect these points with a straight line extending across the graph.
Remember, each linear equation will graphically output a line that only needs two points to be perfectly drawn due to its constant slope. If graphed accurately, the line represents all solutions of the equation, reflecting situations where the linear model applies in real life, from calculating costs to projecting trajectories.