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Find the Standard Form Equation of a Line with Gradient $-\\frac{1}{5}$ and Y-Intercept $(0,-1)$
Mathematics
Grade 9 (Junior High School)
Question Content
Write down the equation of the straight line with a gradient of $-\\frac{1}{5}$ that intersects the y axis at $(0, -1)$. Give the equation in the form $ax + by = c$ where $a, b$ and $c$ are integers in their lowest terms.
Correct Answer
$x + 5y = -5$
Detailed Solution Steps
1
Step 1: Recall the slope-intercept form of a straight line, which is $y = mx + b$, where $m$ is the gradient (slope) and $b$ is the y-intercept value. Here, $m = -\\frac{1}{5}$ and $b = -1$.
2
Step 2: Substitute the values into the slope-intercept form: $y = -\\frac{1}{5}x - 1$.
3
Step 3: Convert the equation to the standard form $ax + by = c$. Multiply every term by 5 to eliminate the fraction: $5y = -x - 5$.
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Step 4: Rearrange the terms to get all variable terms on the left and the constant on the right: $x + 5y = -5$. All coefficients are integers in their lowest terms, so this is the final equation.
Knowledge Points Involved
1
Slope-Intercept Form of a Straight Line
The slope-intercept form is $y = mx + b$, where $m$ represents the gradient (steepness) of the line, and $b$ represents the y-coordinate of the point where the line crosses the y-axis (y-intercept). It is used to quickly write a line's equation when the gradient and y-intercept are known.
2
Standard Form of a Straight Line
The standard form is $ax + by = c$, where $a$, $b$, and $c$ are integers with no common factors other than 1, and $a$ is typically non-negative. This form is useful for solving systems of linear equations and graphing lines using intercepts.
3
Converting Between Linear Equation Forms
To convert from slope-intercept to standard form, first eliminate any fractions by multiplying all terms by the denominator of the slope. Then rearrange terms so that all variable terms are on the left side of the equation and the constant term is on the right, ensuring coefficients are integers in simplest form.
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