Problem 12 - Elementary Algebra

Question

For the following problems, graph the equations. $$ 2 x+5 y=10 $$

Step-by-Step Solution

Answer

Answer: The x-intercept of the equation is 5, meaning the point (5, 0) is on the graph, and the y-intercept is 2, meaning the point (0, 2) is on the graph. The graph of the equation 2x + 5y = 10 is a straight line that passes through these points on a coordinate plane.

Step 1: Find the x-intercept of the equation

To find the x-intercept, set y = 0 and solve for x: $$ 2x + 5(0) = 10 \\ 2x = 10 \\ x = \frac{10}{2} \\ x = 5 $$ The x-intercept is 5, which means the point (5, 0) is on the graph.

Step 2: Find the y-intercept of the equation

To find the y-intercept, set x = 0 and solve for y: $$ 2(0) + 5y = 10 \\ 5y = 10 \\ y = \frac{10}{5} \\ y = 2 $$ The y-intercept is 2, which means the point (0, 2) is on the graph.

Step 3: Plot the x and y-intercepts on a coordinate plane

Plot the points (5, 0) and (0, 2) on a coordinate plane. These points represent the intersections of the graph with the x and y-axis, respectively.

Step 4: Draw a straight line through the intercepts

Now that we have our x and y-intercepts plotted, draw a straight line that passes through both points (5, 0) and (0, 2). This line represents the graph of the equation 2x + 5y = 10. The graph of the equation 2x + 5y = 10 is a straight line that passes through the points (5, 0) and (0, 2) on a coordinate plane.

Key Concepts

The x-intercept

The x-intercept is the point where a graph crosses the x-axis. At this point, the value of y is always zero. To find the x-intercept of an equation like \(2x + 5y = 10\), we set \(y = 0\) and solve for \(x\). By substituting \(y = 0\) into the equation, we simplify to \(2x = 10\). From here, dividing both sides by 2 results in \(x = 5\).
This calculation tells us that the x-intercept is at the point \((5, 0)\). Remember, the x-intercept gives us valuable information about where the graph intersects the x-axis, which is useful for sketching the equation on the coordinate plane.

• Set \(y = 0\) in the equation.
• Solve for \(x\) to find the x-intercept.
• In our example, the x-intercept is \((5, 0)\).

The y-intercept

The y-intercept is the point where the graph intersects the y-axis. At this position, the value of x is zero. To determine the y-intercept for the equation \(2x + 5y = 10\), set \(x = 0\) and solve for \(y\).
Substitute \(x = 0\) into the equation, which yields \(5y = 10\). By dividing both sides by 5, we find \(y = 2\). This indicates that the y-intercept is at the point \((0, 2)\).
Knowing the y-intercept helps us understand where the graph crosses the y-axis, an essential detail for drawing the graph accurately.

• Set \(x = 0\) in the equation.
• Resolve to find \(y\), which gives the y-intercept.
• For this equation, the y-intercept is \((0, 2)\).

The coordinate plane

The coordinate plane is a two-dimensional surface on which we can graph equations. It consists of two perpendicular number lines intersecting at a point called the origin. The horizontal line is the x-axis, and the vertical line is the y-axis.
Each point on the plane is identified by a pair of numbers known as coordinates, written as \((x, y)\). For the equation \(2x + 5y = 10\), we have the intercepts at \((5, 0)\) and \((0, 2)\).
Plotting these intercepts on the coordinate plane allows us to draw straight lines through them, helping us to visualize and understand the graph of the equation. This visualization shows how the values of x and y relate to each other within the equation.

• The x-axis is the horizontal line; the y-axis is the vertical line.
• The origin is the point \((0, 0)\) where the axes intersect.
• Points are represented by coordinates \((x, y)\).
• Graphing the intercepts helps in getting a clear picture of the equation.