Problem 31 - Algebra 2

Question

Find the \(x\)-intercept and the \(y\)-intercept of the graph of each equation. Then graph the equation. \(2 x+5 y-10=0\)

Step-by-Step Solution

Answer

x-intercept: (5, 0); y-intercept: (0, 2); graph is a line through these points.

Step 1: Find the x-intercept

To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\).Substitute \(y = 0\) into the equation:\[ 2x + 5(0) - 10 = 0 \]Simplify the equation:\[ 2x - 10 = 0 \]Add 10 to both sides:\[ 2x = 10 \]Divide by 2:\[ x = 5 \]Hence, the x-intercept is at \((5, 0)\).

Step 2: Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation and solve for \(y\).Substitute \(x = 0\) into the equation:\[ 2(0) + 5y - 10 = 0 \]Simplify the equation:\[ 5y - 10 = 0 \]Add 10 to both sides:\[ 5y = 10 \]Divide by 5:\[ y = 2 \]Hence, the y-intercept is at \((0, 2)\).

Step 3: Graph the equation

Plot the x-intercept \((5, 0)\) and the y-intercept \((0, 2)\) on the coordinate plane. Draw a straight line through these points to represent the graph of the equation \(2x + 5y - 10 = 0\). The line will intersect the x-axis at \(5\) and the y-axis at \(2\).

Key Concepts

x-intercept and y-intercept

Finding intercepts is an essential skill in understanding linear equations and how they graph. The x-intercept is where the graph crosses the x-axis, meaning the y-value is zero. To find it, set \(y = 0\) in the equation and solve for \(x\). For example, using \(2x + 5y - 10 = 0\), substitute \(y = 0\), which results in \(2x = 10\), giving \(x = 5\). Thus, the x-intercept is \((5, 0)\).

The y-intercept, conversely, is where the graph crosses the y-axis, meaning the x-value is zero. To find it, set \(x = 0\) in the equation and solve for \(y\). Using the same equation, substitute \(x = 0\), which leads to \(5y = 10\), resulting in \(y = 2\). Hence, the y-intercept is \((0, 2)\).

These intercepts are crucial because they provide two points that help us accurately sketch the graph of the equation on a coordinate plane.

linear equations

Linear equations form straight lines when graphed and take the general form \(Ax + By = C\). Here, \(A\), \(B\), and \(C\) are constants, while \(x\) and \(y\) are variables. In linear equations, the relationship between \(x\) and \(y\) remains consistent - for every unit increase in \(x\), there is a consistent increase or decrease in the value of \(y\).

Additionally, linear equations have a constant slope, meaning the rate of change between any two points on the line is the same. Understanding this, our example equation, \(2x + 5y - 10 = 0\), can be converted into slope-intercept form \(y = mx + b\), where \(m\) represents the slope and \(b\) the y-intercept.

• A positive slope indicates the line rises from left to right.
• A negative slope indicates a line falling from left to right.

Recognizing these attributes helps in predicting the behavior of the graph based on the equation's form.

graphing equations

Graphing equations, especially linear ones, is a visualization method to understand equations better. The first step is often to locate the intercepts, which provide anchor points on the graph. They allow you to draw a line and understand the equation's relationship visually.

Once you've determined the x-intercept and y-intercept, plot these on a coordinate plane:

• Plot the x-intercept (in our example, \((5, 0)\)).
• Plot the y-intercept (for our example, \((0, 2)\)).

Next, simply connect these points.
This line represents all solutions to the equation, forming a straight line as long as the equation is linear. This method helps students understand the possible values of \(x\) and \(y\) that satisfy the equation, allowing a deeper comprehension of algebraic principles. Graphing is not just a means to an end but a tool to offer insights into mathematical relationships.