Question

The lines \(y=2 x+3\) and \(y=a x+5\) are parallel if \(a=\) ____.

Short answer

2

Step-by-step solution

Understand the slope of a line

Identify the slope of the given lines. The slope-intercept form of a line is given as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Identify the slope of the first line

The equation of the first line is \(y = 2x + 3\). Comparing this to the slope-intercept form \(y = mx + b\), the slope \(m\) is 2.

Describe parallel lines

Parallel lines have identical slopes. This means if two lines are parallel, their slopes are equal: \(m_1 = m_2\).

Set the slopes equal

The slope of the second line \(y = ax + 5\) is \(a\). According to the property of parallel lines, this slope should equal the slope of the first line. Therefore, \(a = 2\).

Key concepts

slope-intercept form

The slope-intercept form is a way of writing the equation of a straight line. This form is written as: \( y = mx + b \) where * \( y \) is the y-coordinate of a point on the line, * \( x \) is the x-coordinate of a point on the line,* \( m \) is the slope of the line, and* \( b \) is the y-intercept, the point where the line crosses the y-axis.
Understanding the slope-intercept form makes it easier to graph the line and to see important characteristics of the line immediately. By identifying \( m \) (slope) and \( b \) (y-intercept), you can quickly draw the line on a coordinate plane. The slope \( m \) tells you how steep the line is and in which direction it tilts. The y-intercept \( b \) tells you where the line intersects the y-axis.

For example, in the equation \( y = 2x + 3 \), the slope \( m \) is 2, and the y-intercept \( b \) is 3.

line equations

Line equations are mathematical expressions that describe the relationship between the coordinates of points on a line. One of the most common forms is the slope-intercept form we discussed earlier: \( y = mx + b \).There are other forms as well, such as:

• Point-Slope Form: \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a specific point on the line.
• Standard Form: \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants.

Each form serves a different purpose, but they all represent the same idea: to show the relationship between the x and y coordinates of every point on the line. For instance, the line equation \( y = 2x + 3 \) can be written in standard form as \( 2x - y = -3 \), which can be useful for different types of calculations and graphing methods.

Knowing various forms of line equations helps you switch between methods for more flexibility in solving math problems.

linear equations

Linear equations are algebraic expressions that create straight lines when graphically represented on a coordinate plane. The simplest form of a linear equation is \( y = mx + b \), which is the slope-intercept form. Linear equations can have one or more variables, but the highest power of the variable(s) is always 1.

Let's break it down: A linear equation in one variable looks like \( ax + b = 0 \), while a linear equation in two variables takes the form \( Ax + By = C \). Both forms will yield straight lines in their respective graphs. More complicated systems of linear equations, which involve multiple linear equations, are used to find the point of intersection of lines, among other uses.
Understanding this helps you realize why the solution to our original problem involves recognizing parallelism through identical slopes. If the lines \( y = 2x + 3 \) and \( y = ax + 5 \) are parallel, it means their slopes must be the same, thus \( a \) must equal 2.

Linear equations are fundamental in algebra and are widely used in fields like physics, engineering, economics, and statistics, as they model real-world relationships in a straightforward way.