Problem 26 - Algebra 1

Question

Write the equation in standard form with integer coefficients. $$y=-0.4 x+1.2$$

Step-by-Step Solution

Answer

The equation of the line in standard form with integer coefficients is \(4x + 10y = 12\).

Step 1: Identify the Slope and Y-intercept

In the given equation \(y=-0.4 x+1.2\), the slope 'm' is -0.4 and the y-intercept 'b' is 1.2.

Step 2: Remove Decimal Coefficients

To convert the slope and the y-intercept into integers, multiply all terms of the equation by 10. This gives \(10y=-4x+12\). In this form, we can see the slope is -4 and the y-intercept is 12.

Step 3: Arrange in Standard Form

The standard form of the equation of a line is Ax + By = C, where A, B, C are integers. Arrange the terms so that x and y are on the left-hand side and the constant value is on the right-hand side. This gives us \(4x + 10y = 12\).

Key Concepts

Linear Equation

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. It is called 'linear' because it graphically represents a straight line on a Cartesian coordinate plane. When you see the equation in the form of \( y = mx + b \), you're looking at a line's blueprint. Here, 'm' represents the slope that tells you how steep the line is, and 'b' is the y-intercept, indicating where the line crosses the y-axis.
Linear equations are crucial for understanding relationships and trends because they predict future values based on past data. They are straightforward because their graph is a line, not a curve or a complex shape.
When solving a problem that involves converting an equation into standard form, you often start with a linear equation in slope-intercept form.

Integer Coefficients

Integer coefficients are important in the simplification and arrangement of equations, especially when writing them in standard form. Coefficients are the numbers directly multiplied with variables in an equation. For an equation to have integer coefficients, these numbers must be whole numbers without fractions or decimals.
In our original example, the equation \( y = -0.4x + 1.2 \) uses decimal coefficients: -0.4 and 1.2. To convert these to integers, you can multiply every term in the equation by the same number. This does not change the equality and rearranges the equation into a format that is easier to interpret and comply with the standard form's criterion.
Converting decimals to integers helps in accurate calculations and representation. It makes the equation cleaner and complies with standard algebraic practices.

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most common and useful forms. It is generally written as \( y = mx + b \), where 'm' is the slope of the line, and 'b' is the y-intercept.
The slope 'm' describes how the y-value of a line changes with respect to a change in the x-value. A positive slope means the line ascends from left to right, while a negative slope descends. The y-intercept 'b' shows where the line intersects the y-axis, providing a starting point on the vertical axis.
This form is particularly useful for quickly graphing linear equations because it gives a direct understanding of the line's tilt and where it begins on the graph. When converting from slope-intercept to standard form, make sure to use integer coefficients, organizing the formula to match the \( Ax + By = C \) format, which is another standard way to represent linear equations.