Question

Write the equation in slope-intercept form. Then graph the equation. $$ y-2 x=-7 $$

Short answer

The equation in slope-intercept form is \( y = 2x - 7 \) with a slope of 2 and a y-intercept of -7. The graph of this equation is a straight line that starts from -7 on the y-axis and rises 2 units for every 1 unit it runs to the right.

Step-by-step solution

Convert to slope-intercept form

The given equation is \( y-2x=-7 \). To convert this into slope-intercept form (\( y = mx+b \)), solve the equation for y. Add \(2x\) to both sides of the equation to obtain \( y = 2x - 7 \).

Identify the slope and y-intercept

In the equation \( y = 2x - 7 \), the coefficient of \(x\) is the slope (m), and the constant term is the y-intercept (b). Thus, the slope \(m=2\) and y-intercept \(b=-7\) in this case.

Graph the equation

To graph the equation \( y=2x-7 \), first, plot the y-intercept (b), which is -7, on the y-axis. From this point, use the slope to find the next point. Since the slope is 2, you would go up 2 units (rise) and move right by 1 unit (run). This will give you another point. Draw a straight line that passes through these two points. This line represents the equation.

Key concepts

Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can be written in many forms, including slope-intercept form, which is particularly useful for graphing. This form is expressed as:

\[ y = mx + b \]

where \( m \) is the slope of the line and \( b \) is the y-intercept, which is where the line crosses the y-axis. Solving the exercise involved converting the given equation to this form, which makes it straightforward to identify these key characteristics and visualize them on a graph.

Graphing Linear Equations

Graphing a linear equation involves plotting points on a coordinate plane and connecting them with a straight line. The slope-intercept form facilitates graphing since you can easily plot the y-intercept (\( b \)) on the y-axis. This point is where the line crosses the y-axis. Next, use the slope (\( m \)) which is the 'rise over run' (the amount the line goes up divided by the amount it goes right) to find another point on the line.

Using the slope \( 2 \) from our exercise as an example, we 'rise' up 2 units and 'run' to the right 1 unit from the y-intercept to find a second point. Draw a straight line through these points, and you've successfully graphed the linear equation.

Slope and y-Intercept

The slope of a linear equation represents the steepness or incline of the line and is usually denoted by \( m \). It shows the change in the y-value for each unit change in the x-value along the line. A positive slope means the line rises as it moves from left to right, while a negative slope indicates it falls.

The y-intercept is the point where the line crosses the y-axis, represented by \( b \) in the slope-intercept form. It's crucial when graphing because it provides a starting point on the graph where the line will pass through.

In our exercise, the slope-intercept form \( y = 2x - 7 \) immediately tells us the slope is 2, indicating that for each step right on the x-axis, the line rises by 2 steps. The y-intercept is -7, placing our starting point below the origin at -7 on the y-axis.