Problem 25 - Algebra 1: Concepts and Skills

Question

Write in standard form an equation of the line that passes through the given point and has the given slope. \((3,-2), m=5\)

Step-by-Step Solution

Answer

The standard form of the equation of the line passing through the point (3,-2) with a slope of 5 is -5x + y = -17.

Step 1: Determine the y-intercept

Using the formula \(y = mx + c\), substitute the given point and slope into this equation to determine the y-intercept. The point given is (3, -2), and slope is 5, so the equation becomes -2 = 5*(3) + c. Solving for c gives c = -2 - 5*3 = -17.

Step 2: Write the equation of the line

Now, using the slope and calculated y-intercept, we can substitute into \(y = mx + c\), which gives the equation of the line in slope-intercept form: \(y = 5x - 17\)

Step 3: Convert the equation into standard form

Converting the equation into standard form (Ax + By = C), we move all terms except the constant to the left side of the equation: -5x + y = -17

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a straight line. It's expressed as \( y = mx + c \), where \( m \) represents the slope of the line and \( c \) is the y-intercept, which is where the line crosses the y-axis. This form is particularly useful because it easily showcases the rate of change of y with respect to x, which is the definition of the slope, and the starting value when x is zero, the y-intercept. It simplifies graphing linear equations and understanding their behaviors.

For example, if a problem gives you the slope of a line and a point through which the line passes, you can substitute the point values into the slope-intercept form to find the y-intercept. Once you have both the slope and y-intercept, you can graph the line or use it in calculations, positioning the line in the coordinate system with clarity and ease.

Finding Y-Intercept

To find the y-intercept of a line, you need to know the slope of the line and a point through which it passes. The y-intercept is the value of \( y \) when \( x \) is zero. You can find it by using the slope-intercept equation \( y = mx + c \).

Start with solving for \( c \), by plugging the point's coordinates \((x_1, y_1)\) into the equation and using the given slope \( m \), to get \( y_1 = mx_1 + c \). After arranging the terms, you can determine the value of \( c \) and thus have discovered the point where your line crosses the y-axis.

For instance, in the given exercise, substituting the point (3, -2) and the slope 5 into the equation gives us -2 = 5(3) + c, which simplifies to find the y-intercept \( c = -17 \). This shows the versatility and practicality of the slope-intercept form in not only understanding but also quickly finding crucial characteristics of a line.

Linear Equations

Linear equations form the backbone of algebra and appear as the functions that graph to straight lines. A linear equation in two variables typically looks like \( Ax + By = C \), where A, B, and C are constants, and x and y are variables. This form is known as the standard form of a linear equation.

The beauty of linear equations lies in their predictability and simplicity. They model relationships with a constant rate of change, which is valuable in various scientific, economic, and social contexts.

As part of moving from the slope-intercept form to the standard form, you might need to rearrange terms. For example, the line equation \( y = 5x - 17 \) from our solution converts to standard form by moving all variables to one side of the equation, resulting in \( -5x + y = -17 \). In this form, A equals -5, B equals 1, and C equals -17. The standard form is preferred for certain practical applications, like when establishing constraints in optimization problems or solving systems of equations.