Problem 28
Question
Find the \(y\) -intercept of the line. $$ y=-3 x+7 $$
Step-by-Step Solution
Verified Answer
The \(y\)-intercept of the line is 7.
1Step 1: Identify the equation
The given equation is \(y = -3x + 7\). You need to find the \(y\)-intercept of this line.
2Step 2: Substitution
To find the \(y\)-intercept, set \(x = 0\) and substitute this value into the equation. So, the equation becomes \(y = -3(0) + 7\).
3Step 3: Solve the equation
After substituting, solve the equation: \(y = 0 + 7\).
Key Concepts
Linear EquationsGraphing LinesSolving Equations
Linear Equations
Linear equations form the bedrock of algebra and precalculus, and they offer a multitude of applications across various fields. These equations represent lines in a two-dimensional space and are typically written in the form \(y = mx + b\) or \(ax + by = c\), where \(m\) stands for the slope of the line and \(b\) represents the y-intercept.
The y-intercept is the point where the line crosses the y-axis. It's an insightful feature because it tells us the value of \(y\) when all other variables are set to zero. For instance, in the equation \(y = -3x + 7\), the number 7 is the y-intercept. It implies that when \(x = 0\), \(y\) will be 7, marking the spot on the graph where the line touches the y-axis.
The y-intercept is the point where the line crosses the y-axis. It's an insightful feature because it tells us the value of \(y\) when all other variables are set to zero. For instance, in the equation \(y = -3x + 7\), the number 7 is the y-intercept. It implies that when \(x = 0\), \(y\) will be 7, marking the spot on the graph where the line touches the y-axis.
Graphing Lines
When graphing lines, we translate linear equations onto a coordinate plane as visual representations. This process begins by identifying crucial points, primarily the y-intercept and the x-intercept if available, plus additional points using the equation's slope.
The slope, represented by the variable \(m\) in the slope-intercept form (\(y = mx + b\)), gives the line's steepness and direction. For positive slopes, the line ascends from left to right, while for negative slopes, it descends. The y-intercept \(b\) is where the line intersects the y-axis, and it's found simply by setting \(x = 0\). In our exercise, with \(y = -3x + 7\), the graph would start at the point (0, 7) on the y-axis and fall off at a slope of -3, indicating a steep decline for every step to the right.
The slope, represented by the variable \(m\) in the slope-intercept form (\(y = mx + b\)), gives the line's steepness and direction. For positive slopes, the line ascends from left to right, while for negative slopes, it descends. The y-intercept \(b\) is where the line intersects the y-axis, and it's found simply by setting \(x = 0\). In our exercise, with \(y = -3x + 7\), the graph would start at the point (0, 7) on the y-axis and fall off at a slope of -3, indicating a steep decline for every step to the right.
Solving Equations
The process of solving equations involves finding the values for the variables that make the equation true. For linear equations, this usually means determining the x and y values that will satisfy the equation when substituted back in.
Equations can be straightforward or complex, requiring various steps such as distributing, combining like terms, or applying inverse operations. The equation from our example, \(y = -3x + 7\), is already solved for \(y\), making it easy to plug in values for \(x\) and solve for \(y\). To find the y-intercept, we set \(x = 0\) and solve for \(y\), giving us \(y = 7\). This single-step solution highlights the importance of understanding the structure of equations and underscores the simplicity of solving for y-intercepts in linear equations.
Equations can be straightforward or complex, requiring various steps such as distributing, combining like terms, or applying inverse operations. The equation from our example, \(y = -3x + 7\), is already solved for \(y\), making it easy to plug in values for \(x\) and solve for \(y\). To find the y-intercept, we set \(x = 0\) and solve for \(y\), giving us \(y = 7\). This single-step solution highlights the importance of understanding the structure of equations and underscores the simplicity of solving for y-intercepts in linear equations.