In Exercises 46–49, give the slope and y-intercept of each line whose equation is given. Then graph the line. 2x + 3y + 6 = 0

검증된 단계별 안내

Rewrite the given equation in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. Start by isolating the term with y. Subtract 2x and 6 from both sides: 3y = -2x - 6.

Divide every term in the equation by 3 to solve for y: y = (-2/3)x - 2. Now the equation is in slope-intercept form.

Identify the slope (m) and the y-intercept (b) from the equation y = (-2/3)x - 2. The slope is m = -2/3, and the y-intercept is b = -2.

To graph the line, start by plotting the y-intercept (0, -2) on the coordinate plane. This is the point where the line crosses the y-axis.

Use the slope m = -2/3 to find another point on the line. From the y-intercept, move down 2 units (negative rise) and right 3 units (positive run). Plot this second point, then draw a straight line through the two points to complete the graph.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as y = mx + b, where m represents the slope and b represents the y-intercept. This form is useful for quickly identifying the slope and y-intercept of a line, making it easier to graph the equation. Understanding this format allows students to convert standard forms of equations into slope-intercept form for analysis.

Finding the Slope

The slope of a line indicates its steepness and direction, calculated as the change in y over the change in x (rise over run). In the context of the equation provided, rearranging it into slope-intercept form will reveal the slope directly. A positive slope indicates the line rises from left to right, while a negative slope indicates it falls.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis, represented by the value of y when x equals zero. In the slope-intercept form, this value is directly given as b. Identifying the y-intercept is crucial for graphing the line accurately, as it provides a starting point on the y-axis.