Write an equation (a) in standard form and (b) in slope-intercept form for each line described. through (-1, 4), parallel to x+3y=5

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Identify the slope of the given line by rewriting the equation $ x + 3y = 5 $ in slope-intercept form $ y = mx + b $. To do this, solve for $ y $: subtract $ x $ from both sides and then divide by 3.

Since the new line is parallel to the given line, it will have the same slope $ m $. Use this slope and the given point $ (-1, 4) $ to find the equation of the line.

Use the point-slope form of a line equation: $ y - y_1 = m(x - x_1) $, where $ (x_1, y_1) = (-1, 4) $ and $ m $ is the slope found in step 1.

Simplify the point-slope form equation to get the slope-intercept form $ y = mx + b $ by solving for $ y $. This completes part (b) of the problem.

Convert the slope-intercept form back to standard form $ Ax + By = C $ by moving all terms to one side and ensuring $ A, B, C $ are integers with $ A \geq 0 $. This completes part (a) of the problem.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Standard Form of a Linear Equation

The standard form of a linear equation is written as Ax + By = C, where A, B, and C are integers, and A should be non-negative. This form is useful for identifying intercepts and for certain algebraic manipulations. Converting an equation to this form often involves rearranging terms and clearing fractions.

Slope of a Line and Parallel Lines

The slope of a line measures its steepness and is calculated as the ratio of the change in y to the change in x (rise over run). Lines that are parallel have identical slopes. To find the slope of a line given in standard form, you can rewrite it in slope-intercept form or use the formula -A/B.

Slope-Intercept Form of a Linear Equation

The slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. This form clearly shows the slope and where the line crosses the y-axis, making it easy to graph and understand the line's behavior. It is often derived by solving the equation for y.

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