Question

Find the \(x\) -intercept of the graph of the equation. \(y=0.1 x+0.3\)

Short answer

The x-intercept of the graph of the given equation is -3.

Step-by-step solution

Set y to 0

Since we are finding the x-intercept, we need to set y equal to 0 in the equation. Thus, we have \(0=0.1x+0.3\).

Solve for x

Next, we solve for x in the equation from Step 1. Subtracting 0.3 from both sides of the equation gives \(-0.3 = 0.1x\). To solve for x, we then divide both sides by 0.1, resulting in \(x = -3\).

Key concepts

Linear Equations

Linear equations are the simplest form of equations that you will encounter in algebra. They have one or two variables and when graphed in a coordinate plane, they always form a straight line. That's why they are called 'linear' – they make a line. In the general form, a linear equation looks like this: \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept.

Understanding the components of a linear equation is crucial. The slope, \(m\), indicates how steep the line is, and the y-intercept, \(b\), tells us where the line crosses the y-axis. In the equation \(y=0.1x+0.3\), the slope is 0.1, and the y-intercept is 0.3. It means for every unit you move to the right along the x-axis, you move up 0.1 units along the y-axis. The intercepts are the points where the line crosses the axes, and in particular, the x-intercept is where the graph crosses the x-axis, and y is zero.

Graphing Equations

Graphing equations is a way to visually represent the solutions to those equations. When you graph a linear equation, you are taking the abstract concept of the equation and turning it into something you can see and understand. This is very helpful for finding intercepts, understanding the rate of change, and predicting values within the scope of the graph.

To graph the linear equation \(y=0.1x+0.3\), you start by plotting the y-intercept on the graph at the point (0, 0.3). Then, using the slope, you rise up 0.1 units for every 1 unit you move to the right. You plot another point and draw a line through both points, extending it across the graph. The graph can then be used to identify the x-intercept visually, which would be at the point where the line crosses the x-axis.

Solving for x

Solving for x is the process of finding the value(s) that make an equation true, in which 'x' stands for an unknown value that we aim to determine. When an equation includes the variable 'x', and we are asked to solve for this variable, we perform algebraic manipulations to isolate 'x' on one side of the equation.

Using the equation from the exercise, to solve for x when finding the x-intercept, we set y to zero because at the x-intercept the value of y is always zero since it's the point where the line crosses the x-axis. The given equation becomes \(0=0.1x+0.3\). Then, by subtracting 0.3 from both sides, we get \(-0.3=0.1x\) and dividing by 0.1, we find that \(x = -3\). Therefore, the x-intercept of the equation is at \(x = -3\), which is where the graph of the equation would cross the x-axis.